DirectEnergyConverstion

DirectEnergyConverstion - I92 I’ H UTtJ V O 1.. T A I...

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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 195-1 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'p-typc and a-type 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 erg-seel. 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. wine-Ill represents a minimum energy. Itcan never be a fraction ofthis valne I‘lLIL'iII-Ii“ 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 5|jt‘. 5.2 A RITEVII'LW OI" RADIATION l‘klNC‘lI’Ll-ZS 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. 5-I: t‘, In; :- ftr/A -—- 0.62 X It) '5‘" erg-sec 3 X It)” cm/sec X soon >< arr cm :13] x [It'll-erg, 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 Niall-war _': Nphlftl/Anv- IrWe assume that outside the atmosphere the solar spectrum has an intensity 0 .. . [JO-US 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’I-IOT'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 5-1. 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. 5-1 we plot in arbitrary units an actual spectral distribution as a function of energy for several of the cases listed in Table 5-1. In Fig. 5-2 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 p-n 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 p-n 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/Mcrn-I} X l0“-I 2.5 3.0 3.5 FIG. 5-1. 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 long-wave 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. 5-1: 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. Phonic-undatedvity 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. 5-2. 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 electron-hole 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 retort-tilt?“ time. We may now use an equation such as [it]. 3-63 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 p-rt 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- '= = (5-6) 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. 5-6 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’} (5-7) 'l' ‘l'he term rerunthinun'ou router or trapping router is another name For :I spatiallyr tin-atom!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 u-type 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(‘ UttNI-nn'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 [rt-tr 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. 5-5. 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 n-type region than in; the p-type region. electrons on the n side of the junction diiTuse down the; concentration gradient to the p-type region where they recombine with free: holes. Positive holes will flow at the same time toward the n-type region irig the valence band. in a very short time (on the order ot'a inillimicrosecondlii the charging up process is completed. with the p-type region possessing fll‘l' excess of negative charge and the n-type region taking on an excess of posi-i 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. 5-3. This contact potential causes the; energy levels of the n-type region to be displaced downward and p-EYPB: 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 .iysten-z 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 p-n .lUNCT'l“l{)NS I99 p-type region n-typc 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 p-njttnetion 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 I-ipeaklllg. 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 p-type region can easily flow "down" into the $in I‘Dgion. This gives rise to a thermally generated current i“ which is "Cutly primortioni-d to the number ol‘ thermally excited electrons in the HYDE region upset/11 exp [— (sum — t1r)/U~‘T)i t (5-9) W . . . . he“- lilc p subscript denotes values in the p—type region. In the n-typc 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, mutt-is]. [1‘ these electrons can cross the potential barrier or. am -~ tint-"h they can enter the p-type 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 right-halal side of liq. 5-10 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 p-tync region. so that it is more negative with respect to the a-type region. by applying a so-callctl reverse bag- or retype region n-type region Voltage Source, 1" ‘ Ila——+ t‘ H—u- --— l,. [itcctroncurrents ?_I_ Adi-I—cl/ Potential enerev for electrons li‘lti. 5-400. The dependence ol'the electron generation and rcconthimttion currents across a p-a 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 p-lype region and holes in the a-typc region. sec. 5.3 Sl'iM[CONDUC'I‘URS ANI) p.» ,IoNC'rtoNS 20! pntypc region n-tync region v} -— —' -I- _ m . - + + ._ __ .l. -- , L. lilectron currents Potential energy to: eiectt’ons [.‘IG. 5-4th). 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 n-type region and holes in the p-type region. Ittttgnitutle'l' V. the energy barrier that electrons in the a-type region will see is now All |- vi”, and it is virtually impossihle For any electrons from the a-typc region to surmount this harricr anti enter the p-typc region. This is shown in Fig. 5-4ta]. 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 p-typc 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 tt-si . . . . - - - - 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. (TI-L 5 barrier the electrons in the a-typc 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. (5-H) 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. 5-1] we obtain i. —- I” —- it. esp I‘eV/(irTJI -- i", one tar/o'er — n. (5-12) 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}. (5-13) where i.. is called the saturation or dark current. Equation 5-l3 is sometimes called the rectifier equation. The equivalent circuit of a plmtovoltaic cell can be drawn as shown in Fig. 5-5. 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. 5-5. (a) Ekguivalent circuit of an illuminated p-a 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 ULprivate-lit circuit consists of a constant-current 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. 5-l3. Thus we have i = i. — 1y (5-14) 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. 5-13. Thus. we have .; = J. — J.- (5-16) or J = J. w J.,{e>(p [eV/(it’l‘ll — l}. (5-17) 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. (5-19) 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. 5-l7: Il' [Tiff til-I” inf". [(1 Vow/(k l-llu ] _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‘ til-Imus : IEJVmp/{f‘l‘rll VH‘J'JJN [I ..|_ I. | -|-- eV..,._,,/(it7'l' (5-23} 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/{ lle-iiflne 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. 5-6 a typical voltage-current 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. 5-6. 't'ypieul voltage-current characteristics for a silicon p-OI'I-H 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 electron-hole 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‘l-tU'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 cmwunn-alion 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. 5-7. 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]. 5-35- 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 (5-27) 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. 5-2? and 5-28 we find a relation between the dilTusion constant and the mobility e = it. (5-29) 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 n-type on p-type cell as shown in Fig. 5-8. 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 5-24 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 Whit-tie. 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 electron-hole pairs in the semi- Conductor. We express this symbolically as J. z 1;...“ H r)[l — esp (—uttlcnm. (5-30) 208 l’l-IO’I‘OVULTAIC' manna/trons. ("H_ s / / / Iznet'to’ l'I'om sun n-type contact ' ' s. _T_ I '/ [JAZZ a-type nunerial 4 p-type material Load p-type contacts + I... FIG. 5-8. A cross section ofan a- on p-type 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 electron-hole 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 well-written 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 eireultll|1 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. 5-30 serves as the definition ol‘ the collection cl‘tic-Ieney: 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—to-volLime 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 5-2 we tabulate the (532.) not: = TABLE 5—2. '1‘ 1-1 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 lit-u (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 ivii-melon 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. (5-33) 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. 5-24 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. 5-24 can be Written as m flames. _ Ilium: F‘s-l Nphgnv We can interpret this equation For a silicon cell by noting that in”. is about {Fl-NM, and the voltage at maximum power is about one-third 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 (Tl-21,1. 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.) ' 11-min” 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 (5-36) 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 liletlme-ol 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,., (5-37) 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,. = (5-38) 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 = (peel-1n)”- Colubining Eels. 5-31 5-38, and 5-39 yields Jun.) = 2.23 X lU”"I‘”p,,p,,n,.itTeL,.—' exp [—tiflfUcTJ], (5-40) 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 |’I--l(.)'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. 5-4” and 5-18 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. 5-9. 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 l-L'nergy gap (cV) FIG. 5-9. 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 'mCiL-ney 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. so-ctltlt‘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 5-3 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. S-Itl. The increased absorption ol' the atmosphere resulting l'rom atmospheric gases1 humidity. altd dust tend to deplete the ultraviolet wavelengths [see Figs. 5-1 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 electron-hole 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 5-l l'or various atmospheric condi- “i‘llh' times the elliciencies presented in Fig. 5-10. liven though elliciencics I‘or 1‘ the case ill a- 3 corresponds to a relative Inunidily ol' about St! percent. TABLE 5-3. PERFORIl-IANCEr OF SOLAR ENERGY CONVERTERSI Hm: (percent) V (V) n mAfc J (amp/cm“) Cost: '_ (Dollars_.-"watt) (safe) #1: V-sec a flu cm cmgi'V-sec 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 10-16 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 urn-5 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. 5-20, 5-21, and 5-22 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. (5-41) 54:2 limitations on photovoltaic energy converters. We will briefly dc- :zllbtflsome lol‘ the factors, as detailed in lief-s. [9] and [10]. that prohihit Tab]: ‘Ijotovoltztic converters from achieving-the 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’I-IO'I‘OVOL'I‘AIC GENERATORS. Ct-i_ 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 electron-hole 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 voltage-amperage 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 technique-inllueiiced 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 min-mu in the generation of electron-hole 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. 5-12. 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'elcelron-hoie piers generated under the assumption of the existence clan abrupt absorp- li0i1 edge with complete ahsorption and zero reliection on its high-energy Side. After Woll'e [to] with permission. dill-crent 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'on-liolc 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'tliTiCIu-Ui: incomplete absorption can he'alleviatet'l somewhat by rapidlt-i to materials that have absorption coeliicients that iiicreasc‘very it With photon eiici gy. Figure 5-13 shows the absorption characteristics 218 l’I-lt)'l'U\r‘0l.'l‘Alt‘ (BliiNl-ERA'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. 5-13. 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 electron-11‘?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 Ell-2M[(.'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 electron-hole. 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 electron-hole pan' generation in a getlllcmttltlelt'lt' With an coergylgap ol {1‘} eV, asstlllltllg that all photons wtth mime-at energy to create pairs are actually ahsorla-tl. This hunt: and the number ol‘clcetron-Iiole pairs generated by sunlight per square centimeter ol‘ exposed area Ipet‘lse'eon'd as a lunctam ol energy-gap {as given in Fig. ‘53) represent basic hmltatlons and are completely Independent ol technique factors. (4-) A signilieant number ol'the electron-hole 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 built-in 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 steady-state 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. 5-18. The energy required to generate an electron-hole 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 electron-hole pairs were generated; this number can never be reached by photon absorption l‘rom direct sunlight. The so«cttlled voltage l'actor (VJ-1), 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. 5-18, by the energy gap ol' the material (in volts) t“yield VQF. == iris/ti” -—‘ int?‘/(et'.fl)_] In [fill/J” + |_]. (5—42) hLlL caltulz-Itlons 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 voltage-current 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. S-l-l 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. 5-14. 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- (5-43}; SEO 5.5 THt-z 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. 5-17, which we have con- vet-ted from a current density equation to a current equation. I = 1.. — 1.. {exp [e V/(kTfl —« 1}. (5-45) Combining Eqs. 5-44 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... (5-46) 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 Sit-percent 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? ~— ltt-t- 50C 1.; : Inn—156E: all « ‘-' It)!” enr” acceptors in the p-ltlyet' net poWer output. In earlier discussions we noted that it the layer all M; : It)” cnr“ donors in the n-laycr 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]. 5-37: “all? _ my... Jrl i ll] — L" The density of holes in the n-type region may be found by calcul uct using 5-38. The pa product is pit —‘ nil '—" 2.23 X ll.l"'T” exp(—t-1../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'k-Fjj';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 iii-type 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 p-type region (10” unfll'l. greater than the number of excess electrons in the ire-type region (Ill1T cm‘“): current flow is nearly all carried by holes. because the p-type region is injecting-3. 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. 5-25 to calculate diffusion length for the holes in the ii-type region. lirst calculating the hole diil'usicll constant by means of the Einstein equation m .11.ka 41111 cm" 10 - — " e vol t-see .-- ..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. 5-3? is p.,_a_o_._. {3.12 x ltli‘e_i_~n_'l_)_(_1.otl x lit r-I commutationmay"? t... _ 1.112 x ltl"'~'crn =' 5.06 In; [it"lilainp/cm". Jaw] :' The Ace I‘regi 5.5 'ri-in 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 5-2. 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. 5-33 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 ii-type 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. it-I' =- 2 with energy in excess of the energy gap of Silicon 15, [mm Fig. 5-2. 3 x It)” cm“a see". Thus the short cir- cuit current density is. from Eq. 5—30. (5-34) =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". (5-30) We may now calculate the open circuit voltage from Eq. 5-13: 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. 5-20: exp learnt/(RTHU _l' (“Imp/(k-Tj} = .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 __ lit-IE!" 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 = 6-54 X in '3 WEIilHi’JCI‘l'li‘l. ill \\ ( J \ a The photon densin arriving at the cell surface is found from Table 5-1 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. -. Burl-ace “ 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 5-35 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 p-type 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 n-type layer superimposed on a base ol'p-type silicon. The a-type layer is made bit dill‘using phosphorus into the surface of p-type 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. 5-15) 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 a-type layer. This allows a reduction in a-type layer thickness while simultaneously obtaining a decrease in series resistance. I . The grid structure as illustrated in Fig. 5-15 consists of line metal strips ".1 contact with the a-type layer separated by a suitable distance. It is acct-5531'?! 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. 5-15. (a) Standard (far left) and nonstandard gridded configurations of solar cells. (b) One way of providing large area contacts to the p-a 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 a-type material layer 2' units thick (as Illustrated tn Fig. 5-3) 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“) " = lit-é x ll.l”cm""'} X {1.60 x l(}"“'coulombflllli'cmi/volt-schfl ' = (1.125 ohm—cm. “the t , , t . thall‘tt-hwc have again assumed half the donor states to be tilled. We also assume is lh‘ “ll-Woe 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 7-3 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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DirectEnergyConverstion - I92 I’ H UTtJ V O 1.. T A I...

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