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Unformatted text preview: I92 I’ H UTtJ V O 1.. T A I C" (i If N If. R A 'I‘ [J R bi. (_" H ‘ 3 noted that a voltage was developer] when light was directed onto one u" um
electrodes in an electrolyte solution. The cll‘eet was Iirst observed in a sum
in INT? by W. (1. Adams and R. 15. Day. who conducted experiments with
selenium. Other early workers with solids included Schottky. Langu‘ and
(_irontlahL who did pioneering work in producing photovoltaic cells wilh
selenium and cuprous oxide. This work led to the developn'lent of‘ mm“!
electric exposure meters. In 1951 researol'iers [tuned to the problem ol‘utiliz.
ing the plmtovoltaic effect as a source of power. [a that year several groups‘
including workers at the RC‘A and Bell Telephone Laboratories. achieved
conversion ellicicncics of about (1 percent by means ofjunctions ol'ptypc
and atype scmieonduetru's. These early junctions. commonly called p...”
junctions. were made of cadmium sulfide and silicon. Later workers in the
area have achieved elliciencies near 15 percent by using improved silicon
p a junctions. A review of developments in this field has been given by
Babe III. 5.2 A REVIEW OF RADIATION PRINCIPLES We introduced in (‘11. 3 the id at that energy is not indefinitely indivisible.
We stated that. the smallest piece of energy that collld be transferred in a
process was called a quantum: that light is a form ofcnergy and a. quantum
of light we termed a pardon. I’Ianck suggested that the energy of a photon
is proportioned to the frequency of the radiation associated with it. t'. to. (5—1] where t'. is the energy of the Photon. ft is Planck‘s constant [ﬂ 6.6."! X I047
ergseel. and a is the frequency in cycles per second. Because we have asse
ciated a Frequency with the radiation. giving, it a wave characteristic and
at the same time have stated that energy is transferred in discrete tll'ltttttttlti.
giving it a ctn'puscular UI‘tttl‘llClL’l‘iHIic‘ we say that light has a dual nature “
not explainablc solely in terms of either waves or particles. I I We note that [it]. 5] states that at any given frequency which intPl'c'“
any given wavelength. since they are related by r'  A» where c is the HP‘fud
of' propagation radiation energy is always in whole multiples of In. wineIll
represents a minimum energy. Itcan never be a fraction ofthis valne I‘lLIL'iIIIi“
a photon of a given wavelength could not have less. We therefore consider
radiation from the sun or any other source of radiation as a stream “I‘Unurﬁ:
chunks called photons. each one carrying an energy esaetly equal 10 l‘
frequency times Planck‘s constant. . . a
('onsidcr a beam of red light With a wavelength A hilllt) A. W“ m I 5jt‘. 5.2 A RITEVII'LW OI" RADIATION l‘klNC‘lI’LlZS I93 find its energy in electron volts [using 3 X It)“' cm per sec for the speed of
light) by a direct application of liq. 5I: t‘, In; : ftr/A — 0.62 X It) '5‘" ergsec 3 X It)” cm/sec X soon >< arr cm :13] x [It'llerg,
but by definition
I electron volt t'eV)  1.60 X [0 11“joule 2 1.60 x If} "‘’ erg
so
__ 3.3] X If) "I erg
' " 1.60 x to v erg/(CV) It follows that the energy of a photon with a wavelength of 3000 A would
have an energy of a little more than 4 CV. Now we may consider the energy source with which we must work in
utilizing plu'ttovoltaic devices. The spectral distribution of sunlight depends
on many factors. including the three sources of atmospheric absorption.
namely (a) atmospheric gases (0th N3. and so on); (b) water vapor: and (c)
dust. Each of these absorptirm mechanisms tends to deplete the ultraviolet
in a preferential malmcr. The effect of these sources of absorption can be
described by means of an optical path length or through which the light
passes. and by means of the number of centimeters of precipitablc water
vapor w in the atmosphere. We delinc m by the relation in 2 l/cos :: where
z is the angle between the line drawn through the observer and the zenith
and the line through the observer and the sun. In the course of a day : varies
from t)[I' deg to a minimum, 2m“, that occurs at noon; :...;.. also is a function
of the season of the year between the limits 2min = latitude :t: 23.5“ deg.
The simplest case is, of course. when 2...“, — U and then m = l. The parlour/fax is a quantity useful in solar cell calculations; it is de
fined as the number of pitt'JIt'H'IS crossing a unit area perpendicular to the
light beam per second. If we let rt: denote the intensity of the light in watts
Per square centimeter. then the number of photons carrying that energy Np,
may be computed quite simply if we make an assumption concerning the
aft‘l'tlgc energy of each photon. That is II" :: Natl“ 2 Niallwar _': Nphlftl/Anv IrWe assume that outside the atmosphere the solar spectrum has an intensity 0 .. . [JOUS watt per cm and that each photon carries on the average 1.48 eV,
“11 We may calcnlate the photon llux as  2.08 eV. ' l eV 'onlc
N” __ I ‘5 walls I J
H 01' cn'1'‘ X L48 CV X I.(‘10>( It} '“Iioule watt~see '— 5.8 X Ill'ltcranee} ". I94 l’IIOT'OVOLTAIC GENERATORS. CH 5 Table 5! gives some indication of the variation in solar intensity a
the photon density for various values of m and w. The total number or 501”
photons Np. as given by Table 5[ is computed by summing the number at .
photons in the energy range from zero up to the maximum energy (Rhoof
4 eV) found in the solar spectrum. ut TABLE 51. PARAMETERS OF THE SOLAR SPECTRUM
AS A FUNCTION OF ABSORPTION CONDITIONS “It 4‘ 8o Np?" m w Comments (Wg’em'l) (CV) (Nu/seeming.
o 0 Outside atmosphere 0.135 1.48 5.3 l 0 Sea level, sun at zenith 0.!06 1.32 5.0 x 10w
2 0 Sea level. sun at 60 deg from zenith 0.088 1.28 4.3 x we
3 0 Sea level, sun at 70.5 deg From zenith 0.075 1.2] 3.9 x 10:1.
1 2 About 50 %_relativc humidity 0.l03 1.25 4.3 x 1011
3 5 Extreme condition 0.059 1.18 3.2 x 1011'
I 0 Cloudy day (7000°K Black Body) 0.012 1.44 5.2 x In"! In Fig. 51 we plot in arbitrary units an actual spectral distribution as a
function of energy for several of the cases listed in Table 51. In Fig. 52
show how these same conditions will affect the photon density as a function
of the cutoff energy. That is, we plot the number of photons in the sol_
distribution per unit area per unit time whose energy exceeds the energy gap;
of the material. 5.3 OPTICAL EFFECTS IN SEMICONDUCTORS
AND pH". JUNCTIONS To set the stage for considering the physical process of turning light in"?
electricity in a pn junction it will be helpful to consider several 0th”.
Optical phenomena in semiconductors. It is interesting to note that man. "
semiconductors that exhibit unusual properties under electrical and tlierm_ '
excitation also exhibit interesting properties when irradiated with Gleam.
magnetic waves of various frequencies. For example, silicon. 3 Natalia}
that has found wide application in transistors, appears to have the WP“: r;
metallic luster when viewed with ordinary light. When viewed undfﬁr long:
wavelength infrared radiation, however, silicon becomes transparent “w H EEC" 5.3 SEMICONDUCTORS AND pn JUNCTIONS I95 m t}, n' = 0 m  l. n' — 0
(sea level) but terms. ath'tttaq‘} “l1. “£35513 “um m1 3. n! = 5 [Energy (eV)
1.238 2.4% Intensity per u
M
D “.5 Lil 1.5 2.0
l/McrnI} X l0“I 2.5 3.0 3.5 FIG. 51. The solar spectrum intensity per unit wave number as a function
of the reciprocal ol' the wavelength {the energy) for various conditions.
are: Loferski [2] with permission. view a specimen of silicon with very longwave infrared radiation and slowly
reduce the wavelength of the radiation, we note that the transparency of the
specimen increases. Further reduction in wavelength brings us to a point
When: the specimen‘s opaqueness increases abruptly. The shorter wavelength
radiation is of higher energy (recall Eq. 51: 8 = in: = tic/A) and thus when
lhe radiation has a wavelength of HUB microns (lt'h‘l meter =  micron) it
illill'rt15ptilitlﬁ to an energy ol‘ 1.12 eV, which is the forbidden hand width of
Ellison. Therefore, photons of this energy or greater are capable of exciting
Selectrcns from near the top of the valence band into energy levels near
File bottom of the conduction band. This transition is accomplished at the
expense of‘ the photon‘s energy. which is absorbed during the process and
makes the crystal opaque to this frequency of radiation. From this intro
Irflciiltﬁl‘l we see that radiation experiments are capable ol‘ telling us quite a
muralut the band structure ol scnnconductors. We will now explore brlelly
phmoc more semiconductor properties which are usclul to understanding
(inversion devices.
Phonicundatedvity is generally deﬁned as the increase in electrical con IdUe‘ ' . .
“Vlly ot a semiconductor clement when radiation 01" the proper frequency l96 ’ll()'[‘t)Vt)l.'l‘Al(‘ til'tNliRA'l‘URS, Q” 5 5 3
:T 2
t5
5 1X Milt}:
n e
E V
. 3_ 1X ltlll't _l .. J ii 0.5 1.0 l.5 2.0
(5,, tcvl FIG. 52. The potential number of absorbed photons as a Function of the
cutotl energy (the width ol‘ the t'orbiddcn energy band). Alter Lot'erski [:2]
with permission. is directed onto the semiconductor. The property has been widely exploited
in devices designed to detect r:.diation. Photoeondactivity occurs appreci
ably whenever radiation having, an energy greater titan the forbidden energy
gap of the specimen is used to irradiatc the semiconductt‘ir. We describe the
phenomena of photocondactivity l'ollovvil'ig the method ol‘ Azm'oll' ILJJ. When radiation is allowed to fall on a semiconductor and producesf
electronhole pairs per ern”sec. we denote the increase in the density 01 electrons and holes by the following two equations:
do r'—' fir}? where the symbol r": denotes the eti‘eetive lit'ctime ol‘ the carriers and tilt:
subscripts denote whether or not we are describing electrons or holes. WC
should note that this ri" is not the same quantity as the mean retorttilt?“
time. We may now use an equation such as [it]. 363 to describe the. chaotic
in electrical conductivity: dram. WM :30 "r amino” lv gym”) stints.  «rites. (5.4) The contribution of electrons and holes to the increase in conductith
Inily be a t‘unetion of their relative lifetimes or their mobilities or both. i” Spy. 5.3 St'tMIconoUcroRs AND prt JUNt.‘.‘t'toNs I97 measure as with accuracy it is necessary that it be ol" the same order
0f magnitude as the conductivity ol. the semiconductor without radiation.
Thus crystals that are l'airly good insulators (having relatively low dark
conductivities). in general. make good phott‘ieontluetors. For example pure
or compensated crystals ol' cadmitu'n sullide (CdS) are essentially insulators
in the dark having conductivities of about it) 1“ (ohm—em) '1. When illumi
nated with light ol' a l'retpiency that corresponds to an energy that is the
width ol' the crystal‘s Forbidden band. the conductivity may increase many
l'old. Cadmium sullide and cadmium telluritle have similar characteristics
in this respect. The photocttrrent 1,. that is produced by illumination ol‘ a Semicon
Lillt‘ltil‘ that is an insulator in the dark may be expressed as a function of
the following variables: F", the total number ot'electrons and holes produced
each second by the absorbed photons; the ell'ective lil‘etime; and the transit
time it, = eF’a"‘/T,., (5.5) where 'I'}. is the transit time— the time spent by an electron in moving
between the two electrodes connected to the semiconductor. It may be
determined from the interetectrotte spacing and the mean tlrit't veiocity ol'
the electrons to be 7i '= = (56)
where It]. is the drift mobility and E is the applied electric ﬁeld. By expressing
the electric tield as the applied voltage divided by the interclectrode spacing,
We write liq. 56 as '— I‘ll
_ “V.
The larger the cll'ectivc lifetime 7*} the greater will be the photocurrcnt l',,.
ll‘wc imagine that the lil'etinte of an electron is greater than the lit'etimc ol'
:1 hole. this means that holes are trapped quickly in recombination eenters‘l’
while the free electron esists long enough to be swept out of the crystal 3’} (57) 'l' ‘l'he term rerunthinun'ou router or trapping router is another name For :I spatiallyr
tinatom!purrin .vtrth" such as those created by donor and acceptor impurities. dislocations,
littet'stitial atoms. and so l‘orth. As tin example ol' the role these states play. consider a
localized energy state lying far above the top of the valence band in an utype semicon
llLtetor. 'l'hesc states are normally empty but there eststs a linile probability that a tree
ulsctron may transfer from :1 state in the conduction band to one ot'these lowu‘dying states.
lit't‘ttuse these states are at a lower energy. the transition by the electron is accompanied
it an L’ltLEI‘Ey release either in the torn: ol'a photon or a phonon. The crystal imperfection
‘5 thus a "trap" for electrons. Simihu'ly. a paype specimen can contain Emir man. The term
reconIbination center is generally reserved l'or an itt'lpet'lbction that has a high probability
“l' Capturing an electron I’rtun the. conduction band and then losing the electron to the
Valence htttttL thus "capturing, ll hole"; recombination centers are generally located near
[it L'cntcr’ ol' the forbidden bantl. tea I’HU'I‘OVU.'I'r‘\l(‘ UttNInn't‘olts. (TH .5.
' .'..l by the applied licld. Since charge neutrality must be preserved. the negativ.
.ef electrode injects another electron until the Free electron can recombine with
a hole. Thus a photon may appear to cause more than one electron to h made available [rttr conduction. This apparent gain is denoted by a mi:
factor 6 1 r," (sis which is a direct index to the clliciency ot'a photoeondaetor. as can be seen:
by substituting G into liq. 55. The gain factor can be increased by decreasin'
the transit time through the interelectrode spacing. It may also be incl—
hy increasing the cll'ectivc lifetime.
velocity. the number of sites
of trapping centers. The plu'itoctmversitui device that has
the n n junction. A p n junction can be made during the growing of .;
crystal, For example. by suddenly adding an excess ol‘ donor impurities to
the melt than which the solidified crystal is being drawn (see Section 3.832
on the growth of scmicmuluctor crystals). ' Since the density of electrons is larger in the ntype region than in;
the ptype region. electrons on the n side of the junction diiTuse down the;
concentration gradient to the ptype region where they recombine with free:
holes. Positive holes will flow at the same time toward the ntype region irig
the valence band. in a very short time (on the order ot'a inillimicrosecondlii
the charging up process is completed. with the ptype region possessing ﬂl‘l'
excess of negative charge and the ntype region taking on an excess of posii
live charge. A contact potential difference having an energy At; develops,
across the junction ol‘such a magnitude to just oppose the Further ﬂow of"
electrons and holes due to the concentration gradient. The region adjacent t0:
thejunction is said to contain a space charge and is sometimes referred t0 as
a transition region. The extent ol‘ this region around the junction is ltJ“"’l0'
10"“ cm wide. The encrgydcvel diagram corresponding to this ﬁnal cquihbs
rium condition is illustrated in Fig. 53. This contact potential causes the;
energy levels of the ntype region to be displaced downward and pEYPB:
region to he shifted upward so that the Fermi levels of the two regions‘
remain horizontal and continuous at thejunction. We generalize this 03531:"
vation Further by restatng a condition that had been previously assumcd.‘._
in a .iystenz in titermoi eqniiiirrinnt the Fermi ievei energy is constant titt'ottgilﬁfii'
the entire system. This is analogous to two containers ﬁlled with a llqmd connected by a pipe in which the liquid seeks the same level in both 60“;
tainers so long as nothing disturbs the system. it should also be noted “1:.
this contact potential cannot be used to deliver power to an external lea ':
As soon as we introduce connecting leads to the pwnjunction. we intl'OduGe ‘ ‘ I case
which is mverscly related to the drit'
available for trapping. and the el'l‘ectivenes' attained the highest elliciency _: .. I 5513C 5Ij SEMICONDUCTORS AND pn .lUNCT'l“l{)NS I99 ptype region ntypc region rt acceptor
it hole it donor e  electron iiv 6 [pi L"
Fern'ti A 6 _ test); :L— Density of
donors and
acceptors 1316.56. A pnjttnetion schematic showing thejunction, the energy hand
:iCth'tc. and the density of donors and acceptors as a Function of position
in thejunction. New Contact potentials; this brings the total circuit voltage to zero as long as
fall Parts of the system are kept at the same temperature. Thermtu'lynamically
Iipeaklllg. We say that a contact potential is totally unavailable in a system
in thermal equilibrium.
I The small number of thermally excited electrons that take up residence
in “10 cond action hand ol‘ the ptype region can easily flow "down" into the $in I‘Dgion. This gives rise to a thermally generated current i“ which is
"Cutly primortionid to the number ol‘ thermally excited electrons in the
HYDE region upset/11 exp [— (sum — t1r)/U~‘T)i t (59) W . . . .
he“ lilc p subscript denotes values in the p—type region. In the ntypc 2.0" l’ H O "r 0 V 0 LT A I (‘ (ii If". N E R A 'l" O R S . C' H. 5 region. the ntnuher ol' thermally excited electrons in the conduction band
is given by ti“ a: .‘tl esp (cm, muttis].
[1‘ these electrons can cross the potential barrier or. am ~ tint"h they
can enter the ptype region to reeomhinc with the holes. This produces
recombination current i, flowing to the lel'l that is oroi‘iorlional to it AI esp [ —(At:  am“, — snitch]
at] esp [vistas  than  than  lirl/lki’jj
' "ll mill "'(liattrl ' ' il't'lt'lll‘liyrli (5'40) We note that the righthalal side of liq. 510 is exactly the same as liq. 5.t)_
That is. the potential harricr adjusts itself" to such a value that at equilibrium
the current flowing to the right is the same as the current ﬂowing to the
left. or .t', in. We continue our exploration ol‘ the p a junction by perforating some
simple cspcrintcnts. ll‘ we hias the ptync region. so that it is more negative
with respect to the atype region. by applying a socallctl reverse bag or retype region ntype region Voltage Source, 1" ‘ Ila——+ t‘ H—u — l,. [itcctroncurrents ?_I_ AdiI—cl/ Potential enerev
for electrons li‘lti. 5400. The dependence ol'the electron generation and rcconthimttion
currents across a pa junction with a reverse hias. The reverse saturation
current is title to the New ol‘ particles that are minority carriers in the re
gions from which they come. This reverse current is titlU entirely to tiitl'u
sion resulting from the mintu’ily particle concentratitm gradients at tutti
near the junction and thus is not tlepentlent on the applied voltage. In thls
case the saturation current is Line to electrons in the plype region and holes
in the atypc region. sec. 5.3 Sl'iM[CONDUC'I‘URS ANI) p.» ,IoNC'rtoNS 20! pntypc region ntync region v} — —'
I _ m . 
+ + ._ __
.l.  , L. lilectron currents Potential energy
to: eiectt’ons [.‘IG. 54th). The dependence ol‘thc electron generation and recombination
currents across a (M: junction With it Forward hias. The Forward current
across a junction is due to the majority carriers in the regions l‘rom which
they come. 'l‘ltey cross the junction because of the ctmcentrtttion gradient
from one sitle of the barrier to the other. The electric ﬁeld influences these
gradients so as to litvor a Llill'usion current in the Forward direction. the
forward current in this case is LlUt: to electrons in the ntype region and
holes in the ptype region. Ittttgnitutle'l' V. the energy barrier that electrons in the atype region will see
is now All  vi”, and it is virtually impossihle For any electrons from the
atypc region to surmount this harricr anti enter the ptypc region. This is
shown in Fig. 54ta]. Thus the recomhination current for electrons !, is very
small. This hiasing does not alI‘cct the generation current For electrons If, to
“HY L‘Xtent. hecause the ntlnthcr ot‘ thermally escitetl electrons in the ,n—type
1"‘ltion is not changetl. The reverse bias also limits the nurnher ol' holes in
the valence halal that can go l'rotn the p sitlc to the a side. Recall that holes.
unlike electrons. prcl'er to go “Uphill.” We thus have about the same i“ Ellltl
*1 reduced i. l'or holes. ’ ll ‘v'r'efia'u'ard tries our p a junction. we get the situation illustrated in
Flg' 54(5). Once again we have Llone nothing to change the another ol'
lllvrntally cscitetl electrons in the ptypc region ill'ltl thus the current i“ is not
“lllt'tstetl. However. the applied voltage in this situation reduces the energy l “5’ convention we normally assume the voltage l' to he less than zero when the ttsi . . . .    
1"?" “t “L't.’.'tttve. and we assume. l' to he. greater than zero when the {Millie is positive.
at. i‘ \fl'ltcn we apply a reverse hlas. the potential barrier that electrons on the n sitlt: see Is vi") en:  at". 202 l‘ll()'l‘()\t0l.'I‘Al(‘. (il'iNt'iRATORS. (TIL 5 barrier the electrons in the atypc region ol' the conduction band see and
tints increases i. for electrons according to the Boltzmann distribution law
by a l‘actor exp tel’ftl'r'i’ﬂ. Because i“ does not change and tr equals [a at
equilibrium we may write fr, :—.— 1.. exp :cV/(tc’i‘l'l. (5H) The net electron current that will llow in the circuit is the ditl'erencc
between the two currents i, and in; thus l'rom liq. 51] we obtain i. — I” — it. esp I‘eV/(irTJI  i", one tar/o'er — n. (512) This current is zero when V 2 (land increases to large values for positive [3V‘
and decreases when eV is negative toward a negative saturatirm value ﬁt“. The hole current flowing. across the junction behaves similarly. The
applied voltage that lowers the height of the barrier for electrons also lowers
it for holes. so that large numbers ol‘ holes flow from the .0 region to the
a region under the same voltage conditions that produce large electron cur.
rents in the opposite direction. We note that electron and hole currean
going in dilTerent directions add; the total current. including the ell‘eets of
both holes and electrons. is given by i. : Mexp [eV/(ir'i'll — 1}. (513) where i.. is called the saturation or dark current. Equation 5l3 is sometimes
called the rectiﬁer equation. The equivalent circuit of a plmtovoltaic cell can be drawn as shown in
Fig. 55. The elaborate realistic model of Fig. 5—5(a) is generally replaced it.. r..T
‘
/ Junction
Constant current Constant current
source source
it!) (b)
FIG. 55. (a) Ekguivalent circuit of an illuminated pa junction photo» voltaic cell showing the internal series and shunt resistance. (it) The sixth
piilicd equivalent circuit. that is used in this chapter. EEC, 5.3 SILMICONDUC'I‘ORS AN!) s” JUNC'I‘HJNS 203 with the sinlplilied circuit shown in Fig. 5—50)). For our purposes the simpli
HL‘LI circuit. is perfectly adequate and yields essentially the same result as the
more realistic model. Because the operation of a photovoltaic converter
involves the microscopic action previously detailed. we describe its opera
tion in terms of a macroscopic device that yields an equivalent result. The
ULprivatelit circuit consists of a constantcurrent generator delivering a car
Wm i. into a network of impedanees. which include the nonlinear impedance
of lheliunetion Ry. an intrinsic series resistance it” an intrinsic shunt resist»
ance RM. and the load resistance R... Application ol~ Kirchholl"s law to the simplified circuit will yield several
interesting relationships. When a p njunction is illuminated. the light causes
acnrrelit l' to [low in the load; the magnitude of this current is the dili'erence
betwuen the current that would ﬂow it" the junction were short circuited i.
and the current that flows across thcjunction i}. which we found in Eq. 5l3.
Thus we have i = i. — 1y (514)
or
i = i. — i..{exp [UV/(ICTJJ — I}. (545) where i; is the total current due to both electron and hole flow across the junction, i., is the dark or saturation current, and V is the voltage across the junction. We will now make an analysis of this device as an energy converter. in
so doing we will try to lind those characteristics that lead to high ell‘ieiency
devices. Since it is nu'Jre common to analyze photovoltaic converters in terms
ofcurrent densities. rather than in terms ol'eurrents. we now switch to that
notation. The current densities are based on the area ol' exposed junctions.
The current density that flows through the load. J. is the dill'erence between
the current that would llow it' thejunction were short eircuited. J... and the
current that flows across the junction. J}. which was dclincd by liq. 513.
Thus. we have .; = J. — J. (516)
or J = J. w J.,{e>(p [eV/(it’l‘ll — l}. (517)
The maximum voltage that we could measure on the cell would occur
under open circuit conditions. J r t]. which is
V... = [iii/e) ln {J../J.. l l} (543)
T0 lind the voltage that will produce maximum power density we compute
the power output ol' the device
i’ — JV r (J. — J.,{esp [eV/(it'T'Jl — 1})V. (519)
Taking the derivative of this equation with respect to V and setting the
rcsutt equal to zero yields an implicit equation tor the voltage that maximizes
“1‘: power gamers? l'l '_1 204 l’ll()'l‘(JVUl.'l'Al(‘ (iliNliRA'l‘URS. (1”. 3 exp [blﬁupftt’r't'llll  chin/(tr?) l  J../'J..  esp 'ch./Ur'l'). (5.20}
The current density that maximizes thc power may be t'ound by mm
hitting the cspression tor the maximum power voltage as given by liq.
with the expression for the current as given by liq. 5l7:
Il' [Tiff tilI” inf". [(1 Vow/(k lllu ] _i“ J”
I .i. cVW/(lr't‘) .1"
The maximum power density is then simply
lPlans “ Jenni/mp 5~20 Jam _ , ‘ I ,.I.r Li. . r.
.r is'ﬁiu {is}: .' I’.  i.» _L' ($21) Ul‘ tilImus : IEJVmp/{f‘l‘rll VH‘J'JJN [I .._ I.
  eV..,._,,/(it7'l' (523} The power density input to the junction is simply the total number or
plmtnns in the solar spectrum. NM, times the average energy of each at" those photons. #1“... Since the dark current density at, is usually live or more
orders ol‘ magnitude smaller than the short circuit current density J", Wu
may approximate the maximum ellicicney of the converter as m, lit” VH1 to“ "I" l VIII inI
ll Hm,“ \.' . . 1 I . I
J I _l_ u Vmp/{ lleiiﬂne We should note that the number ol'pl'totons with energy greater than 6,
decreases as t'.“ increases, while the ratio .l,./J.. and consequently, me in.
creases with t1“: it is evident that a...“ will pass through a maximum as a
Function of ti“. In our analysis We have assumed that the internal shunt resistance Ra.
is m ueh greater than the load resistance Rm and the internal series resistance
R, is much less than R... Whereas the first condition, which causes most of
the j unctiun current to be delivered to the load. is easy to achieve. creatinga
small series resistance is a more diliieult task. The larger litrr is. the greater is
the power that is dissipated in thejunction and the smaller the voltage across
the load. We now examine in Fig. 56 a typical voltagecurrent plot for an actual
solar cell; we show the ell‘ect that load resistance has on the power yield
ol‘ the cell. The nonlinear characteristic ol' the junction is clearly evidcﬂt
in this plot. 5.4 l‘ROl‘liJR'l'llilS DESIRED IN
SEMICONDUCTORS FOR CELL USE . . . . . . #1
In the previous section We considered the mechanism by winch :1 F;
junction converts radiation energy into electrical energy. We also tch‘chd ‘1 (5.24) . 51': C. 5 . 4 S li M I CO N I) U ("I'D R S lit) R. (.1 F. L l.. Ll S Isl 205 so  .I 
l Illumination 1;“ Load
it resistance 70 _ level IZS mW cm"! / Mtlsill'lurn power point 8 ltltl mW cm“ Ln
:3 4t} 30' Output current. mil'ﬁampcrcs asim um / i‘l UWL‘I' etnnglc .  Output voltage. volts FIG. 56. 't'ypieul voltagecurrent characteristics for a silicon pOI'IH cell.
The open circuit voltage depends logaritlnnieally on the illumination.
while the short circuit current is a linear l'unction ol' the illumination. The
actual voltage to the load is independent of cell area and is a function only
of load resistance and illun'iination level; however the output current de—
pends on the illumination level, load resistance. and cell area. miDression For the efﬁciency ol' a plmtnvoltiac device in terms ol' such macro
SUUI‘ic parameters as dark current density, short circuit current density, and
voltage at the maximum power point. These expressions. while uscl'ul l‘or
50ml: tasks in cell design, tell us nothing about the microscopic properties
“‘5” Wnllltl be desirable in the semiconductor of which the cell is made. In
“"5 section we will indicate what those properties are and which materials
“1"” Proved successful in photovoltaic cell use. We begin by considering, some ol" the ramificatitms ol. the series resist
ance iIllt'oduced in the circuit analysis ol' the. last section. To achieve a high
:SFyvelrsion clliciency it is desirable to produce electronhole pairs within a short distance ol the p n Junction. Inlcctrons and holes produced lar
mm umjunction simply recombine without contributing to the cell‘s output. 206 l‘ltU'I'UVUI.'I‘AIt' tittNIv'nA'l'URS. ('.'ll. 5. The average distance a carrier tlil't'uses het‘ore recomhination is called the ' rifﬂhs'tml fetter}: and We now consider what it means and how it is calculated If we generate excess carriers in a small region ol'n semiconductor. they
land to tlill‘nse away from their point ol'generalion hccause ol'lhc conceer
lion gradient: they tlill‘tISe down the concentriltion gradient away lil't‘tl‘t] the
high cmwunnalion at the point of generation. These esccss carriers I1:va a
ﬁnite lit‘etinie. r"‘. so that they will ultimately disappear hy reconihinatiom
The “WI“8.; dimmer; a carrier dill‘uses hel‘ore reeonthimllion is called lite
tlitl‘usion length L. and is related to the dilTusion constant t) and the Iil‘etimc by
L = (Dri‘lmu (5—25) Wt. "my interpret the dill'usion constant. physically. by considering a
semiconductor that has a concentration gradient of carriers as is shovvn in
Fig. 57. The carriers' motion in response to the gradient is analogous to the  “'11 im ‘lltll"‘
on (\AEAOR A AZA: A Atcc] ltr t lc gt. l
{‘H ADA A A 0% AA‘CJA (__) 0 Hole (charge l) lt'lG. 5—7. By doping a specimen so that. it has a concentration gradient of
acceptors. an electrostatic lieltl can he set up along the specimen. The elec
trostatic field is set up hecause the higher hole concentration at the right
hand causes more holes to drift from right to tell than from left to right. Inixing‘t of Mo dil'l'crcnt gases or liquids. The electric. current density that 11
concentration gradient will produce in the .t‘ direction is simply (In _
.1“ = —eD . (5 26)
(IX
. . . . . . . I y.  . u
1 It two tlltlercut gases or haunts are allowud to remain Ill contact. then. Wlli llllvu
tll'adual mixing ot' the two. even in the nhsenee ol’convcclion currents. 'l‘hese processes
mixing are tlescrihetl by t‘iek’s laws: the [irst one has a form identical to In]. 535 sl:(‘_ 5.4 SEEM[CONDUCTORS FOR CELL. USI". 207 where the minus sign occurs because dill'usion occurs toward the region of
d.crc.sitig concentration. Il'this tlill'usion produces an electric field it”. steady
rm; conditions are reached when the conduction curretit.J...t = arE. ts equal
( l. .
:0 the ditl'nston current Jo.
Jami E Jr}
e 1":
Heels = —eD (527) Th9 presence ol‘ the field E will cause a potential energy dill'crencc ol' n1agni«
tuck; (stair to exist over a distance x. Thus. it" the carrier concentration is low
6.101.th so that we may use Boltzmann statistics to describe the energy
distribution of the carriers. then the ecuiecntration gradient as a I’tlnetion at
x is _ a = C exp [—eEx/(lel. (5—28)
where C is a constant. Upon combining Eqs. 52? and 528 we ﬁnd a relation
between the dilTusion constant and the mobility e = it. (529)
which is known as the Einstein reterion. This equation is quite useful in de
termining D because u can be easily determined from Hall cchct experi
ments. How does the dill'nsion length relate to the series resistance with which
this discussion began? As we stated earlier, only those carriers produced
near tltcjunetion (10“ cm < L < 10"“ cm) contribute to the output at the
cell. For this reason cells are generally Fabricated with the junction located
very near the surface so that thejunetion intercepts the maximum amount
ofineittcnt light. In the ease of an ntype on ptype cell as shown in Fig. 58.
the u~type layer is about 10—" cm thick. This very thin layer. through which
the load current ntust ﬂow, is the origin ol' the series resistance R. that we
mentioned earlier. The thickness ol‘ the n~typc layer therel'ore must be a
compromise between the value ol' It... and the collection elliciency ol' the
Junction l‘or photocxeitcd electrons and holes. E(Illation 524 reveals that the ellieiency ol‘ a photovoltaic energy con
‘I'Fi‘tcr is linearly dependent on the short circuit current density J... The short
Circuit current density is proportional to the clliciency with which the
csrrters generated in the bulk are collected and delivered to the external
CII‘Cuil, the l'raetion ol' the incoming photons ahsorhed in the cli‘ective
Whittie. the l'raction ol‘ the radiation not transmitted completely through the
Junction. and the num her ol'photons per second per unit area ol'p n_iunction
whose energy is great enough to generate electronhole pairs in the semi
Conductor. We express this symbolically as J. z 1;...“ H r)[l — esp (—uttlcnm. (530) 208 l’lIO’I‘OVULTAIC' manna/trons. ("H_ s / / / Iznet'to’ l'I'om sun ntype contact ' ' s. _T_ I
'/ [JAZZ atype nunerial 4 ptype material Load ptype contacts + I... FIG. 58. A cross section ofan a on ptype material photovoltaiejunetion. where am, is the collection cilieieney. r is the reﬂection eoellicient, esp (—al)
is the l'raction ol' the radiation transmitted. or. is the absorptitm eoellieient. l'
is the thickness ol‘ the absorbing semiermduetor. and up. is the number of
photons per second per unit area whose. energy is great enough to generate
electronhole pairs. The mathenmtitsll description ol‘ the absorptimi ol‘ light by a solid is
very similar to the description ol' the attenuation of gamma rays. X rays. or
molecular beams, topics that are discussed in any wellwritten modern
physics book such as Blanchard e! at. [4; Photons are removed l‘rom the
incident beam by collisions that prod uee cscited states or free electrons and
holes. Il'ds is the original light intensity ol‘ a beam and II, is the intensity alter"
traversing a thickness )5 ol‘ the solid. then III = [II] ux}(—wx}‘ where a is the absorption eoellleicnt. It may be demonstrated that the ah! I.
sorption eoellieient is inversely related to the mean l'ree path. Since CKI1i‘i‘di':
is the l'raction transmitted through Lhejunclitm. then [I — esp (mlli is the:
Fraction ol‘ the original beam that has been absorbed at the distance tr ‘ I
We note that not all the minority carriers that are generated in the 50nd
by the radiation contribute to the power developed in the load. because 50111.?
of them recombine With majority carriers either inside the bulk of ill“ 5‘1"“
conduetor or at the surl‘aee. We have recognized this by delining the cellist:
tion cllicieney a...” as the ratio ol' the carriers passing through the eireultll1 EEC. 5.31 SEMICONDUC’I‘URS tron can. USE 209 Shun; circuit current. density, J,) to the total number of carriers generated in
the solid per unit time. Rearrangement of liq. 530 serves as the deﬁnition ol‘
the collection cl‘ticIeney: J..
(l — rill — esp(—m’}']en,,;.' W has been computed l'or‘the simple geometry case of an infinite plane
_” junction at x =— t: It was lound to be a l'unction ol' the absorption con
stant For the radiation or, the minority carrier lil‘etime 7* (the lil'etimc ol‘ a
hop; in an retype region or the lifetime of an electron in a p~type region),
and the surl'ace recombination velocity s. The surl'ace recombination velocity
is the ell‘ective velocity at which all minority carriers appear to he swept into
the sarl'aee where they disappear in surl'ace trap energy levels. Surl'ace
recombination is evaluated by measurement ol‘ the total recombination rate
as a Function ol'surl'aee—tovolLime ratio. The surface recombinalitm velocity
is very sensitive to surface treatment: high surl'aee rectunbination velocities
are found for ground or sandblasted surfaces whereas low surface velocities
occur in surl‘aees polished by chemical etching. In Table 52 we tabulate the (532.) not: = TABLE 5—2. '1‘ 11 I"? C 0 L [.4 E (.1 'l‘ l O N I": F l7 I C I I": N C Y at”.
AS A [illNC'l’l 0N Ii: 0!" s, L, A N I.) or [T 1: LY litu
(em/see) (cm) (em ")
(I ll] 10“ 0.6!
U 10”” It}:i 0.47
a 10"“ It)" 0.65
ltlt) Ill 'i' It.)3 0.6]
w 10'2 It):l [1.25
w 1U":1 ltl" 0.23
:0 ll] '” It)" U.t)tll
VJ ll} "' It)"L (1 X [EH 1“ From Rel". 1 With {assumed equal to Ill"?l em. Collection el'licieney as a l'unetion ol' the sarl'ace recombination velocity. the
LlIllusion length L. and the absorption cocllieient or.
a hiiii"illItljnat'ion ol'ble 5:21 reveals that the most Favorable condition l‘or
3”“th iviimelon etheleney that Jor which l/L <3: I and tor which the
adviwhutttUn‘thlnat1on velocity Is low. fhese ealeulattmis indicate. the
umcienldllit 0 chemically etching the surlaee ol pl'iotovoltale cells. ‘For
cc” he {:1 th‘li'ptton 1t is necessary that the thickness ol the top sarlaee ol 'the
1% {'w‘ ll‘II‘onanately equal to the reciprocal ol‘the absta'ptlon constant,
i has It) “ < .’ < 10"" cm. The ltletlme ol the minority earrlers can 210 I‘HO’I'OVOLTAIC GENERATORS, CHIS be determined if we assume that the thick ness of the top surface l is equalt
0.] of the diil'usion length, L = (Dr’l‘)'r‘=. Then if D = 10 cm2 per sec. tho
lifetime of the minority carriers should lie in the range [0"i’ see > Ts:
[0—1" see. It can be shown [5] that s will not reduce the collection ell'icienc
if.r/D<< [/L and 3/13 << a. was r = 0.1L = or" for (x in the range gum, 9’
Table 5~2, both ol‘these conditions are satisﬁed its is less than 10"cm pm Se:
These modest requirements on 7* and x predict that many materials Should
be capable of yielding high values of m", and the collection ellicieney should
not infect the choice of the best materials for solar conversion. The magnitude of reﬂection losSes can be estimated by using the rela,
tion between the index of refraction 7 and the reflection cocliicient r, namerI r = n — lr/(v + Ir. (533) Because 7 has not been measured for many materials that are of interest in
photovoltaic devices, an empirical relation due to Moss [6] is used to csti~
mate the index of refraction 8,,('}/)‘1 m 173 (5.34} that has been found to be approximately true for materials of the zinc blend
or diamond lattice structure. If 8,F = lcV, then 7 = 3.6 and r = 0.3L ll
(1,, = 2 W, then 7 = 3.05 and r = 0.26. We conclude that materials that
have energy gaps that make them of interest for use in photovoltaic energy
converters will all have about the same reﬂection loss. We are now in a position to further simplify the ellicieney expression
given by Eq. 524 and to ﬁnd an upper bound for the ellicieney at which a
cell can operate. This may he done by writing the short circuit current as
J. = Kins. where K is a constant that includes all the losses we have just
discussed, namely, reﬂection, transmission, and collection. If we assume that
Km 1 and that eV,,,,./kT is much greater than unity, then Eq. 524 can be
Written as m flames. _
Ilium: F‘sl Nphgnv We can interpret this equation For a silicon cell by noting that in”. is about
{FlNM, and the voltage at maximum power is about onethird the energy oltllc
average impinging photons. The results of these approximations yield E111
upper limit on the ellieicncy of a silicon cell of about 22 percent. The last parameter that we will consider involving the microscopic
properties of semiconductors is the saturation current density J... Shockiclr
[7] has presented a detailed analysis or this problem: we will give a sirnplt‘tr
picture of the events in the neighborhood of the junction. We lirst assume that thejunction region is thin in comparison with U15
dilTusion length and so recombination within the junction proper miIY h“
neglected. We consider a heavily doped p region in contact with a moderath 55C. (1017“ H
J“. Thus. 5.4 SEMICONDUC‘I'URS t‘oa (Tl21,1. use 2] region which is u I'avorable condition For achieving a low value of
we assume that the current is carried solely by the holes generated
, I L. H region that dill'usc over to the ,0 region. The current carried by these
I” ll '5 equal approximately to the absolute value ol‘ the electronic charge. ilL‘h‘ ' , _ , . . I . . H _ ht] mqu the equilibrium hole density, p,” in then legion. tunes the mean
3.) ' 11min” velocity a,” which we assume to be the dill'usion length l.,, ——
iii) rll'” divided by the hole lil'etimc r?” thus yielding
.1‘ l _ (Prelim _ Us u. a (536) Fr}: — L."
1,1 gcmianium at room temperature yaltics of 200 cntpet‘ sec‘i'or holes having
alii'ctime ol' 10“L see are typical. in silleon, holes having a liletlmeol 10"" sec
rich] :1 mean dill'usion velocity ol [0" cm per sec. The hole generation current
density is then approximately Jun.) = p,.eD,,/L,. 2 p..e,.lr.'f'/L,., (537)
where We have used the Einstein relation D = pK'ch. We recall from liq.
344 that the product ol‘ the hole and electron densities in a given region
must be a constant
of _ 2.23 X 10"”7'" exp [—tiy/(ltTll
“I” h it” ‘ p,. = (538)
where n. = equilibrium density of electrons in the intrinsic semiconductor
anti an is the equilibrium density ol'eleetrons in then region. The latter quan
tity may be expressed in terms of the electrical resistivity (as was done in
Ch. 3) as “a = (peel1n)” Colubining Eels. 531 538, and 539 yields
Jun.) = 2.23 X lU”"I‘”p,,p,,n,.itTeL,.—' exp [—tiﬂfUcTJ], (540) where the results are in amperes per square centimeter. Multiplication by the
area ol'thcjunetion converts the answer to l,,. Other methods of calculating
J.. are outlined by Lolerski Tile strong temperature dependence oI'J., is
uDlltlrcut I'rom liq. 5~4U. Keeping in mind that the lower J,. is, the greater is
the ellicicncy, it is apparent why the lower the temperature at which the
solar converter is maintained, the better its perl‘ormance will be. The only other quantities needed to evaluate the ellieicncy and the
saturation current are the density ol' photons that have energy exceeding the
energy gap oi" the material, the total photon density, and the average energy
“I. “108:: photons; these quantities were given in Section 5.2. 5“ Choice of materials. Based on the principles set down in Section 5.4.
We may now make some statements on the selection ol' materials for use In
Photovoltaic converters. 212 ’Il(.)'l'OVUI.'l'A[C’ (iliNI'iRA’l‘URS, CH 5 Two ctl‘eets take place in semicomlueting materials used in solar mm
verters. First, the number ol‘ photons absorbed with energy greater than tit.c
band gap, up” decreases as the band gap increases. t'l‘his point was ilttistmlud
in Fig. SQ where Wt: plotted the number ot' photons absorbed against um
semiconductor energy gap.) As the band gap increases. the Sittlll'tuiun cu“
rent tlcnsityJ” decreases. thus causing an increase. in the output voltage. Th.”
point hecomes clearer when lids. 54” and 518 are combined. The results or
this operation are messy, but qualitatively we observe that a reduction inJI
which occurs on increasing the energy gap. increases the open circuit voltagrt;
as the log ot‘ the ratio a",./J... The eil'cet at these two t'orces on the maximuln
conversion cllieieney ol'a photovoltaic converter is illustrated in Fig. 59. This
curve was obtained by Lot'erski [S'l using a. set ot' equations similar to those
derived in the preceding section. [a computing J.. a material with silieon‘g
properties was aSsumcd; reflection. recombination. and transmission losses
were ignored in computing J... since it was demonstrated earlier that these
quantities do not vary appreciably among dill'ercnt materials. This curve
gives the maximum tt’ieorericut cllieieney which could be expected under the
assumptirms set down in deriving the governing equations. A small cross on Si CiaAs 30 20 newC t percent} l __ I. l . ___. 0 0.5 1.0 1.5 2.0 2.5
lL'nergy gap (cV) FIG. 59. The maximum solar energy conversion ellieieney as a function
ol‘ the energy gap ot' the semiconductor. The maximum measured ciliciency
For various materials is denoted by a cross at the energy gap ot‘the material.
The curve has been calculated for an idealjunetion outside the atmosphere.
Alter Refs. [8'] and '9]. slug 3.4 SI'ZMlt‘UNDLH'TURS "(}( t'!i.. USIL 2I3 mu “Harpy gap value line ol' each material indicates the ucrrmt’ conversion
'mCiLney which has been achieved in real converters. L We note. that the. curve passes through a broad maximum due to the
opcmtiun ot‘ the previously described opposing Forces. The curve. ltowwer.
rump; values ol' a“...l5 higher than tltal ot' silicon l'or semiconductors with
mch gaps between .l amt 2.3 eY. ‘ I I I I Materials having energy gaps to this range include indium phosphide.
[up (L2? eV): gallium at'sentdc‘ (Ea/ts [LES eV): aluminian antimonide.
A151] (Ho CV]; cadmium telluride. (‘d'l‘e (Li eV): sine telluridc. ZnTc
all Lat); aluminum arsenidc. AlAs (llﬁ cV}; and gallium pliosphidc. (ial’
(7.24 W). This range ot' energy gaps would also ineltnle mixed semicon
dluaors l'ormed by con'lbining materials l'rom the MN and [MN semicon—
duclors in dill‘erent proportions. An example of this type material would he
GHAHIEPI..J. which would have an energy gap between .35 L'V (Ga/\s) and
234 UV ((ial‘). Other materials that have energy gaps that would make them
alinterest l'or converter use are iunetions composed ol' two sentieonduetors.
soctltlt‘tt heteroiauctions such as that between selenium (Set and cadmium
selcnide {(‘dSe} and cadmium sullide {CdS} and eupric sullidc (CuS). To indicate the order ot‘ magnitude ot' some ol' the t‘ltiantit.ies that. are
involved in com paling cllieiencics ot' solar converters. We. have taken results
and data From Ret'. l‘ll in Formulating Table 53 on the performance of" solar
energy converters. It is interesting to note that the theoretical ellieiency actually increases
in some materials as the optical path length Hi and the number ol'eentimetcrs
nl'prceipitable water vapor tt' ir'tcrettse‘l‘ as shown in Fig. SItl. The increased
absorption ol' the atmosphere resulting l'rom atmospheric gases1 humidity.
altd dust tend to deplete the ultraviolet wavelengths [see Figs. 51 and 5%}
thus l'avoring converters with small band gaps; the perl'ornuulcc ol‘ large
band gap materials is severely penalised because an insullieient number of
Photons with energies necessary to create electronhole pairs reaches the
converter. We note finally that the ditlicrenec between silicon and the opti—
mum material is greatest. l'or conversion ol' solar energy outside the. atmos—
nhere. Under these conditions as.“ for silicon is l‘).»'l perecnt‘ whereas at...“
“11' a material with an energy gap ot' rs eV is 34.6 percent. Thus, none ol'the
millt'l'ials discussed here appears to oll‘er improvements ot' as much as a
rilL'ltll' ol' two. The maximum power output under various crnnlitions ot' atmospheric
III“f‘l'l‘ttion are shown in Fig. 5] l. 'l'hese curves may be ohtained by multi
plyii‘t‘. the energy densities given in 'l‘able 5l l'or various atmospheric condi
“i‘llh' times the elliciencies presented in Fig. 510. liven though elliciencics I‘or 1‘ the case ill a 3 corresponds to a relative Inunidily ol' about St! percent. TABLE 53. PERFORIlIANCEr OF SOLAR ENERGY CONVERTERSI Hm:
(percent) V
(V) n mAfc J
(amp/cm“) Cost: '_
(Dollars_."watt) (safe) #1: Vsec a ﬂu
cm cmgi'Vsec 3:
(8V) Almeria! 0.53 5 Theory 5.9 X 10—” 44.
10*5 Exp'tl 15 to 60 10—3 250 1200 1.12 Si 0.73
0.50 51 Theory 10—“
Exp'tl 1.2 1200 _ 250 10—5 45 CdS§ 8.3
21 34 7 X 10—“ 214 In 0 95
0.58
0.84 0.50 39
24
41 2.2 X 10“2 1016 Theory 10—19 Exp'tl 1.45 300 30 10—5 CdTe 21 Theory
Exp‘tl l. 35 3000 600 10"5 800 GaAs 4.2 21 4.? x 10—11 i Illumination taken to be 140 mW‘_.:'c From Ref. [91. + § Bend gap is controlled by width of energy gap in C1135. All other properties are those of Si [9]. _ Bare ceLl SEQ, 5.4 SEMICONDUCTORS FOR CELL USIE 215 _____— m l
n! 2 (with
absorption
hands) 10r m=3
urn5 1.0 1.25 1.50 1.?5 Energy gap (eV) 2.0 FIG. 5—10. Maximum conversion efﬁciency of :1 solar cell as n i'un'ction ol‘
energy gap Ior various atmospheric condilions. After Lol‘erski [2] with
permission. small band gap materials might be higher with atmospheric absorption,
power densities sull'er losses of as much its 50 percent under these conditions.
The load impedance at maximum power can be Found by combining
Eiqs. 520, 521, and 522 derived in Section 5.3. The result of these opera
tions is simply
25:11:? we re; rm: R _ exp [—ve/(km 'x'iozﬁos
mp — ' " _" "' 9 ‘ '1' M_.l)1{ﬁlgl.t; Vol.4, Ive/(kip) F” 4')
Where L. and Vs”, are computed by the methods indicated in the earlier see
lion. The load impedance For maximum 'poWer is seen to be dependent on
the. energy gap through the quantities l2, and V,,,,,. As the energy gap in
creases, the load impedance rises rapidly. (541) 54:2 limitations on photovoltaic energy converters. We will brieﬂy dc :zllbtﬂsome lol‘ the factors, as detailed in liefs. [9] and [10]. that prohihit Tab]: ‘Ijotovoltztic converters from achievingthe ellicreneles predicted ll'l inili'uc .. borne oli these filetors were prewously encountered when we
i d our discussion of semiconductor properties. The IntiJ or factors are:
ill Reflection losses on the surface; 216 I’IIO'I‘OVOL'I‘AIC GENERATORS. Cti_ 5 30
E 25
U
g 20
E m a t /
E 15 ii.J I: 2
E (with absorption
5 bands]
E
a" it) 1.0 I.25 1.50 1.75 2.0
Energy gap (eV) FIG. 5—”. Maximum power from an ideal photovoltaic converter as a
function of energy gap for dilferent absorption conditions. After Loi'erski
[2] with permission. (2) Incomplete absorption; (3) Utilization of only part ol‘ the photon energy for creation of elec
tron—hole pairs; (.4) incomplete collection of electronhole pairs; (5) A Voltage Factor; (6) A curve l'actor related to the operating point at maximum poster; (7) Additional degradation ol' the curve due to internal series resistance. These Factors can he divided into several groups as i'olltiWs: Factors l, 2!
and 4 can he coiiihined and called the overall collection eli'iciency, as was
done earlier; all three ol‘ these Factors are related to the absorption charm?
teristics of the material. Factors I through 4 determine the short circml
current. Factors 5 and 6 are related to voltageamperage characteristics or
the device. Factors t. 4, and T are mainly determined by techniques. and
improvements in these areas may all but eliminate these factors from
consideration. Factors 2. 3, 5. and 6. however, have absolute piiyiiicul
limitations beyond which improvement is not possible. It should be 110‘“
that these basic limitations are also techniqueinllueiiced to a certain exic'1t'
We will brieﬂy discuss each of the seven Factors. (I) Reﬂection losses have been reduced to almost zero, by mean? or
transparent coatings with appropriate thickness and index of refractioﬂ SEC_ 5.4 SEMICONDUCTORS FOR CELL USP. 21'? Losses as low as 3 percent for silicon cells have been reported in the literature
9]. We conclude that these losses are no longer of major consequence in
making improvement in cell periormance. (2) Figure 5—12 shows the energy spectrum ol‘ sunlight at sea level on a
bright clear day. The figure also indicates the maximum amount of energy
minmu in the generation of electronhole pairs in semiconductors with «— I’hoton energy (electron volts) 2.0— 1.14_ 1:1 01‘) Oiti 0.65 lgl l _'.____‘__'_ " 63;: 12's wit—trip)
5.8 X 101" pairs/sec ‘ _, _ £—
.I4L .12 m.—
.t' .10 ;[£iicrgy spectrum ol‘ the sun —— ',., r 1.0? ev (Si)
2.8 X 1017 pairs/sec Ill/7
‘ .08 _rEL,  0.68 ev {CiaSli. Ge)
s/sec Energy density (wattsrcm'l) ‘. m ll; "w 4:. 0.3 0.5 0.7 0.9 1.1 1.3 1.5 {.7319 ii 2.3 Wavelength (microns) —u FIG. 512. The energy spectrum of the sun at sea level on it bright. clear
day. and the parts of this spectrum tlliiiZtliJit: in the generation of electron
holc pairs in semiconductors with energy gaps of 2.25, LOT, and [1.68 eV.
Tcspccliwly. Listed for each ol' these cases is the number ol'elcelronhoie
piers generated under the assumption of the existence clan abrupt absorp
li0i1 edge with complete ahsorption and zero reliection on its highenergy
Side. After Woll'e [to] with permission. dillcrent energy gaps. For every value of the energy gap at eiitol'i' line is
Obtained beyond which the photons possess insufﬁcient energy to create an
clccti'onliolc pair. It is also observed that the smaller the energy gap, the
larger the portion ol' the sun‘s spectrum that can be utilized by the cell.
mmi‘rll‘la pl'tliTiCIuUi: incomplete absorption can he'alleviatet'l somewhat by
rapidlti to materials that have absorption coeliicients that iiicreasc‘very
it With photon eiici gy. Figure 513 shows the absorption characteristics 218 l’Ilt)'l'U\r‘0l.'l‘Alt‘ (BliiNlERA'I‘URS. CH. 5
[Us .__._ . .. . _ ._ _, __ __
'5‘ an
E
3
E
.e
U
55
"5 as
U
C
.9
e
8
ﬁ. 10:
It)  ~———
1.0 1.5 2.0 2.5
Energy liwl (electron volts FIG. 513. The optical absorption eoel'licient as a function of photon en
ergy (lie) in electron volts. Al'ter Iailei'skl [9 thi‘l perlnlsstr:in. ol‘ some semiconductors of interest l’or cell use. A rapidly rising absorption
eoellieient. such as that ol‘GaAs. means that a larger l'raetion ol‘ the carriers
generated by the absorption of solar photons will be within a short distance
of the surface ol‘ incidence. The importance of this l'act can be more fully
appreciated il~ we recall our earlier interpretation of the absorption coellt
cient as the reciprocal ol' the mean l'ree path of the. photon in a Stllllt'l..CEJli'
scquently. the "active region“ of the cell that is, the sum of the dillttston lengths {Lu ~l Ls] on the two sides of the junction ean be smaller. This means that shorter liletimes can be tolerated, tints opening the possibilitl' “f
creating a cell approaching its theoretical maximum. Materials with shortL‘I‘
minority carrier lil'ctimes are generally easier and cheaper to make thrill
materials that have previously been considered l'or eel] use. Furtherntﬂflvls
less material will be needed in any given solar cell since it will require Li
thinner lilrn to insure cll‘eetive absorption of the incident radiation. I [c
(3) Figure 5[2 reveals that a large number of the photons that Will in.
absorbed will have more energy than is needed to create an electron11‘?1
pair. Any energy that an impinging photon has in excess of the ener'gi' ol' the material will contribute to the lattice vibrations of the niaterulI ii is
will eventually be dissipated as heat. Tints, those photons whose energy SFC, 5.4 Ell2M[(.'UNI)U("’I‘URS 't'.‘.IR cinta. use. 2:9 lugs mm the energy gap ol'tlie material do not contribute at all to electron—
hole Pllil' generation and those photons whose energy eseeetls the energy gap
lgl‘the material while el'eatliqz, an electronhole. pair have lllL‘JI' excess energy
distaipated as heat. Woll presentsa curve that shows that as percent ollthe
“1:11;”!!ng solar energy can be utilized Ill electronhole pan' generation in a
getlllcmttltlelt'lt' With an coergylgap ol {1‘} eV, asstlllltllg that all photons wtth
mimeat energy to create pairs are actually ahsorlatl. This hunt: and the
number ol‘clcetronIiole pairs generated by sunlight per square centimeter ol‘
exposed area Ipet‘lse'eon'd as a lunctam ol energygap {as given in Fig. ‘53)
represent basic hmltatlons and are completely Independent ol technique
factors. (4) A signilieant number ol'the electronhole pairs generated by photon
absorption will not be created within the space charge region at the p a
junction; only those pairs created within a dillusion length of' the junction
can be collected and separated by the builtin field. The majority ol' those
created at greater distances l'roru the junction will recombine. causing the.
collection ellieieney to tail below Illtl percent. In Section 5.4 we listed the
major l'aetors ol'whieh the collection cliieieucy is a l'unetion. Woll‘lllll, in a
more elaborate analysis. solves a dill'nsion dill'erential equation for the
steadystate condition to obtain an expression for the overall collection
eilieicncy. (5) The largest recoverable voltage in a photovoltaic cell is the open
circuit voltage given by liq. 518. The energy required to generate an
electronhole pair is equal to the energy gap ol' the material which, on a per
electron basis. can be expressed as a voltage. The open circuit voltage is
always observed to be less than the energy gap voltage For the material of
which the cell is made. There are two reasons l'or this: (a) The barrier height is equal to the maximum forward voltage across
the junction; this is determined by the dill'erence in Fermi levels in the u and
P*lype material on both sides of" thejnnction. The Fermi levels, as discussed
in Ch. 3, are a l'unction ol~ impurity concentration and teml'ierature, and
are normally located within the Forbidden gap, thus causing the barrier
Willa to be less than the energy gap. lb) A voltage equal to the barrier height would he obtained only if"
an extremely large number of electronhole pairs were generated; this
number can never be reached by photon absorption l‘rom direct sunlight. The so«cttlled voltage l'actor (VJ1), a measure of how much ol‘ the cell
Potential is being realized, can be formed by dividing the open circuit
““tssc, as calculated by liq. 518, by the energy gap ol' the material (in volts)
t“yield VQF. == iris/ti” —‘ int?‘/(et'.ﬂ)_] In [ﬁll/J” + _]. (5—42) hLlL caltulzItlons ol the quantities appearing to tlus equation have already
L" Elven in Section 5.4. 220 ‘t()'l'(JV(}I..'t‘AIC‘ (iliNtiRATURS. ('H s. an In Section 5.3 we indicated that the maximum power that term be
obtained than a converter is obtained at the operating point which chutes“
the largest area in the voltagecurrent characteristic curve ol' Fig. 5(,_ Thi
point is delined hy the maximum power voltage and current. VW, and J s
Forming the ratio of the products V...,..t,..,, to Vacs"... we obtain what Woll'[lliilj
calls the curve I'actor (CPI): _‘ Vmp‘ll'l'll” .._. Ja :3 V .ﬂr 'I‘ i” cxp (r too/tr 'l 'J
Vital!“ _ Jill In (Jar/Jr: _l— I J I Both the curve l'actor and the voltage l'actor depend on the saturation
current J... As We outlined earlier. the saturation current is dependent. upon (IF. tlte material properties. In Fig. Sll we plot both the voltage lactor and the. LO Curve l'actor. _ your Jaw
V .J 0.8... m. :I ‘ Voltage lactor a” Von/{Eu
“.6 Characteristic l'actor 0.4"” Vim: Juan
= 6t: J» 0.2 2.0 0.5 1.0 [.5 2.5 Energy gap (cV) IllG. 514. The curve l'actor. voltage l'actor. and characteristic footer for a
solar'encrgy converter as a function of semiconductor energy gap width.
Alter Woll' [Ill] with pct'n‘tission. curve l‘actor and their product. which We call the criteria'rerisn'efoster. All threc factors are slightly dependent on energy gap and increase as the energ}l gap increases. The characteristic factor is pretty much independent of tcclt'
niqtte. being sensitive only to the nature ol‘ the material used in the converter (543}; SEO 5.5 THtz DESIGN OF A CONVERTER 221
ml on top of the junction was reduced in thickness, the collection elli—
r as improved because of the larger photon absorption. This solution. t w ierIC)’ . . . . .
Eowcvcr, conflicts with resistance requirements, because as the layer ts
cdlwctl in thickness, its resistance goes up. The ohmic contacts applied
r to the cell also cause its resistanec to go up. but improvements in the
techniques of applying contacts have caused this resistance to become all but negligible.
The voltage drops around the circutt are y 2 IR" "l— Vlllntl and the current ﬂowing in the circuit is givon by Eq. 517, which we have con
vetted from a current density equation to a current equation. I = 1.. — 1.. {exp [e V/(kTﬂ —« 1}. (545) Combining Eqs. 544 and 5—45 we [ind that the voltage across the load when
we include the internal voltage drop ol~ the cell is 'V..,...i z (JCT/e) in [l i (1,,  !)/l'.,] ~ IR... (546) The output power would obviously be reduced by the presence 01‘ the FR.
dissipation in the generator. Possibilities For reducing series resistance
through the development of improved techniques appear good. 5.5 THE DESIGN 0!? A CONVERTER Now we will consider the methods used in the design ol‘ a photovoltaic
converter. This design will be based on properties of existing materials in
order that the calculations be as realistic as possible. Problem. We wish to design a solar power plant to operate a small radio
used as a community listening center in an underdeveloped section of the world.
The poWer required is 5 watts; this power is stored in u hattery system where it is
Used for 2 or 3 hours in the late afternoon or evening period to energize a transistor
radio. Daytime charging by the solar cell system would be expected to store 20 to
3” Whr. Our design will include ﬁnding the performance parameters for the energy
cDﬁVcI‘ter. We assume that the unit. is energized by the sun under conditions given
rlﬂﬂ’l’g: lea: f Z'whieh correspond to the sun at the zenith with In Sitpercent
5 t. hanatltly. the energy ot the photons and their density are given In table .. m Sqlution. We will assume that the converter units are p—tt junctions made
in blltc'tm with the following properties: (7) it has been l'onnd in actual converters that the series rcsislaﬂcc‘ul s, o I I] W T. “WK
.u . u . . ' ' .‘ .. r ., . l ; g‘tcrlfillc5‘ p. 2' u =*I I.
thcccll can. cause deviation from the ideal voltage current chtrtc ’ the T: _ 4m mwmuﬁcc “N am! CIW/vohmscc
This deviation causes the curve to ﬂatten, resulting in a reduction at w A? ~— lttt 50C 1.; : Inn—156E:
all « ‘' It)!” enr” acceptors in the pltlyet' net poWer output. In earlier discussions we noted that it the layer all M; : It)” cnr“ donors in the nlaycr 222 PH OTOVOL'I‘AIC Ci liiNliR/XTORS. CH I
We begin our work by calculating the reverse saturation cum; .. . .. m d. n
J...” of the holes. utilizing Eat]. 537: “all? _ my...
Jrl i ll] — L" The density of holes in the ntype region may be found by calcul
uct using 538. The pa product is pit —‘ nil '—" 2.23 X ll.l"'T” exp(—t1../k'1’") em“”
a: 2.23 X HF" (3011)“ exp (—1.1 H.026)
— 1.59: X It)!” arr“, “ling [113 p" where we have used It)" [1.1126 eV. Now we assume that the temperature of the device is not high enough 10 id.
ize all the donor electrons but is just high enough to cause the Fermi lech to be:
the donor energy level. Let us examine this assumption by considering the “pre
sion for the concentration of electrons in the conduction band. This number m' be equal to the number of vacant donor levels plus the number of holes in valence band: c y h. # ___ ___..glv“__ _ . . [1_ . I ‘ t —~ exp [(8. — tin/(try _ " l i exp [(8. —8;)/{1’cT)]
1
‘l‘ 2Nu [I — I + exp[.L‘EI'I/E'kFjj';zfg We may simplify this expression by noting that exp li—t'lffiic'l’l] is a very smii
number (about 10"”) and hence the last term may be neglected Since We ha .'
assented E}. m 8.; we ﬁnd that concentration of electrons in the iiitype region “1'! 2 = '1: X “in Cm”.
Thus an. = 1.56 x its" = p.11. x lit”)
,0" = 3.12 X tﬂi'cin”. We repeat again that. since the number of holes in the ptype region (10” unfll'l.
greater than the number of excess electrons in the iretype region (Ill1T cm‘“):
current flow is nearly all carried by holes. because the ptype region is injecting3.
large number of holes into the n~typc region. For this reason we calculate only 1'1
hole contribution to the saturation current. We now use Eq. 525 to calculate diffusion length for the holes in the iitype region. lirst calculating the hole diil'usicll
constant by means of the Einstein equation m .11.ka 41111 cm" 10  —
" e vol tsee .
..I. volt
' . [a u ]_ i u . .
x ll.(illll°K X m] K 1‘ 4W1 ﬂash and the hole dilTusion length is
1'... = (D,.rii)”" = [(10.4 em‘'/see](1tt' sec)?” == [.02 K ill"2 cm.
The saturation current density from Eq. 53? is p.,_a_o_._. {3.12 x ltli‘e_i_~n_'l_)_(_1.otl x lit rI commutationmay"?
t... _ 1.112 x ltl"'~'crn =' 5.06 In; [it"lilainp/cm". Jaw] :' The Ace I‘regi 5.5 'riin DESIGN OF A CONVERTER 223 giSI'E'C'
We now calculate the short circuit current. which requires that we estimate
collection etlieieney. To tlo this we use the data in Table 52. Using an absorp
‘on coellieient ol about 10" and assuming an etched surface for our cell. which
W5 us a low surlace recombination velocny. we ﬁnd a collection etliciency ol‘
0'51‘ since our answer is not too dependent on this number. we do not carry out
me t:mborute calculations required to obtain a better estimate. The rellectlon coelil
cién; is obtained by using the empirical lzq. 5~23 lor the index of relrtielion Swirl" = 1T3
l.1('y)" : 173
7:1“ and Eq. 533 for the relleclion coellicient r 2 ('Y — I)“ : {£4_ 11'."
['r *l~ I)“ (3.54 l 1)“ our earlier discussionlol‘collectiOn etiieiency pointed out a condition that would
produce a high collection elliciency, namely f/L << . Using the previously recom
mended condition of! = 0.1L 2: a“. we ﬁnd that the thickness of the iitype layer
should be i’ : ct" = [1.11: : [1.0111 cm. Thus the fraction of the impinging radia
tion that is transmitted is cam—nil) = exp(—lt]a X 10‘“) = 0.368. The number of photons under conditions in z 1. itI' = 2 with energy in excess of
the energy gap of Silicon 15, [mm Fig. 52. 3 x It)” cm“a see". Thus the short cir
cuit current density is. from Eq. 5—30. (534) =I l'.l.3l . J. = Tina“  rlll — exp(at)]wi..i
=1 0.61“ — 0.31)“ — tl.3?}(1.ﬁll X 10“” eoulonibjttil X it)'1 size"I tint—3) {2.7 X 11'1"“ amp/em". (530)
We may now calculate the open circuit voltage from Eq. 513:
Vim = (kT/e) in Lin/Jr, l 1). (5.13) The factor kT/e is simply 11.026 volt. so ",
I __ y_..
V... = t).t12nln{12.7 x iii“min x 10““ i 1), 'n " am" Where We may drop the one in the parentheses in comparison with the current ratio term. Thus
V... = 0.0261n {2.51 X 10'”) = 0.62 volt. 51"“: We are interested in operating the converter under maximum power "PO ' ' l . . i . . . .
nditiens we must now determine the voltage that permits this operation. This r ‘ . .
equires a trial and error solution to liq. 520: exp learnt/(RTHU _l' (“Imp/(kTj} = .l ‘i‘ I’ll/J“
Lixi) (Vllllt/U.026)(l ‘l' Viirip/UJRG} = x 101". naximum HHSWBI‘ lies between (1.54 and 11.55 volt. We will take (1.54 volt as the from Eq Sp???" vultusc. The maximum power current density can now be found __ litIE!" ti/(k Jinn ; 1 __ eprmykar) 'l‘ {Jo/Jun 224 [’HOTOVOLTAlC GeNEItATORs. CH. 5. SEC’ 5.5 THE DESIGN OF A CONVERTER 225
J _ (l).54)/(U.tl2o_)(12.?_X_ 0""} [:1 __ 5416 }_<___l_(_l;‘_" t —l~ {l}.54)/(U.t)26} 12.? >< tu=I
= 12.1 X Ill" 3 amp/cm". (5.21.)
The maximum power density is simply the product of the maximum power cum
density and maximum power voltage: “‘1
Paul 2 ponytail = x = 654 X in '3 WEIilHi’JCI‘l'li‘l. ill \\ ( J
\ a
The photon densin arriving at the cell surface is found from Table 51 to b _ it ‘\
4.8 X Ill” cm“! sec"1 each one carrying on the average .25 eV. Thus the incidean " ‘
energy density is l \\
t _
. . e ‘ tl \
N....E1.... = 4.8 X lll”cm""sec "' X 1.25 cV X Lott >< ll]‘”'JOUlL:(CV)'l ". [agave \
>< [ watt—sec(joulelr'l 2 9.6 >< 10"twaus/cm. . Burlace “
5 l \ \ . u "‘ " \  ‘ ' ' 'I a ‘ ‘ .l I I I I ‘ l \ i
The maximum cllrctcnty ol out cell in tly now bt. computed by dividing the output. .I \ Back surface
power density by the input powet density. \ \ \ ‘1
Paul X 'I ' ' ‘I \\ §
Than — NM; — >6 =' DblLLnl. 1‘ Grid \ ._ p.{ypc cantact
\
Equation 535 gives a means of estimating the theoretical maximum efﬁciency efai "IYPB Ctltnlitct
photovoltaic converter; we may use this equation to see how close our converter. 3
is coating to the theoretical maximum: {N
a—type contact ptype comact
rival/m. 3 X ill” 0.54
“Hm t: i .._.....___I __ :. )w I . I
” Na. 4s >< In” K [.25 “""“'“ (it) So we see that our converter is operating a long way from its theoretical limit. We now consider the assembly of this array into a unit that will supply the
needed power. Considerable work ltas been devoted to ﬁnding a cell geometry
that maximizes the power output of the cell by minimizing the power dissipated
in the cell itself. Photovoltaic converters made of silicon generally have a thin
ntype layer superimposed on a base ol'ptype silicon. The atype layer is made bit
dill‘using phosphorus into the surface of ptype silicon. The problem of securing
large single crystals of silicon for cell use has been. to a certain extent, alleviated
by the use of current collector grids on the surface of the cell. These grids [illus
trated in Fig. 515) oll'er a means of increasing cell ellicieney by dividing cell strucr
ture into sections about :5 cm wide by 2 cm long instead of the usual I by 2 cm Slit}:
Wolf [to] reports that this causes increases of as much as 20 percent in cell eﬂr
ciencics. The reason for this improvement is that the smaller width of the cell I'EP'
resents in el'Tecl a shorter mean path for the hole current and a lower current dc]?
city in the atype layer. This allows a reduction in atype layer thickness while
simultaneously obtaining a decrease in series resistance. I . The grid structure as illustrated in Fig. 515 consists of line metal strips ".1
contact with the atype layer separated by a suitable distance. It is acct5531'?!
again. to satisfy two conﬂicting requirements: wide grid lines have low result
but decrease the active area of the cell, while narrow grid lines expose most 0
surface to the impinging radiation. but have high resistance. Wolf optimizes
line width and spacing for this type of cell. For the sake of completeness let us calculate the resistance of the thin will?“
surface layer of the cell. FIG. 515. (a) Standard (far left) and nonstandard gridded conﬁgurations
of solar cells. (b) One way of providing large area contacts to the pa
Junction. I Chapin ['12_] has estimated the resistance across the width of a cell L. units
.0113” W units wide, and having either a p or atype material layer 2' units thick (as
Illustrated tn Fig. 53) to be W92. 4L..i of [We may ﬁnd out what this value is for our cell by calculating the resistivity
lte autype layer: Roll 2 P" = learn“) " = lité x ll.l”cm""'} X {1.60 x l(}"“'coulombﬂllli'cmi/voltschfl '
= (1.125 ohm—cm. “the t , , t . thall‘tthwc have again assumed half the donor states to be tilled. We also assume is lh‘ “llWoe layer is it cm Wide and 2cm long and has a depth oi'tl.t)tll cm. which
L rmlibrocal oi the absorptimt coellicienl. Thus “5 >< tl.25
Ra“ ;: I L: ‘
4 X 2 X aunt 73 Ohms. hut, : . .
each cell 0] by 2 cm has a surface resistance of 7.8 ohms. For comparison until}
f till:
grid ...
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