Modeling Concentrator Solar Cells Using Detailed Balance and

Numerical Approaches

Tim Bald

Mentors: Seth M. Hubbard, PhD., John D. Andersen, PhD.

Abstract

The conversion efficiency of single and triple junction solar cells were modeled utilizing adetailed balance theory. Simulations were coded in Matlab. Two methods were used to model thetriple-junction solar cell. First, using a modified single junction model and second using Lagrangemultipliers. Error between the two methods was found to be the smallest at the highest maximumefficiency. Conversion efficiencies of 48.1%, 47.4% and 50.6% were found for 6000K black body,AM0 and AM1.5 spectra respectively. A global Lagrange model was also made and used to modelthe effects of concentration and temperature on a triple junction cell.

Introduction

In the quest for renewable energy sources,mankind has placed great interest and re-sources into the progression of photovoltaics.As this interest has grown, various typesof solar cells have been produced: organic,thin films II-IV, single crystalline silicon,single-junction III-V, and multi-junction III-V cells. Among these types of cells, the high-est recorded efficiency has been produced byconcentrator III-V multi-junction cells in ex-cess of 40% [1]. Concerning high efficiencymulti-junction cells, there are still many ques-tions that remain to be answered. For in-stance: What is the theoretical maximum effi-ciency and how can it be modeled? What ma-terials can be used to obtain such efficiencies?How does concentration affect the maximumefficiency? When sunlight is concentrated ona cell, the cell temperature increases, so onecould ask: How does temperature affect cellefficiency? These and similar questions can beeffectively answered by computational model-ing of the solar cell. It is the effort of this pa-per to explore different methods of modeling

solar cell efficiencies via the detailed balancetheory.

A basic solar cell is comprised of a single p-njunction. An energy band diagram of a p-njunction is shown in Fig. 1. The energy lev-els of atoms that comprise the material of thep-n junction are split. They are split into thevalance band, which is filled with electrons,and the conduction band, which is not filledwith electrons. The bands are separated by anenergy that is characteristic to the semicon-ductor material know as the energy gap, Eg.Light of energy hν is shown to be incident onthe p-n junction. The absorbed photons willexcite electrons from the valance band to theconduction band creating electron-hole pairs.Electrons in the conduction band can flow eas-ily from the p-type region to the n-type re-gion of the p-n junction (right-hand side inFig. 1) while holes in the valance band can-not. The opposite is true for the p-type re-gion (left-hand side in Fig. 1). This asym-metry in charge separation forms an intrinsicelectric field that causes photogenerated holesand electrons to flow in opposite directions

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as shown. This makes it possible to collectelectrons generated by incoming photons toproduce a current. That current can be con-nected to a load to produce a usable power.

Figure 1: p-n junction. White dots repre-sent holes, green are electrons. hν = photonenergy, Ec = lowest unoccupied state, Ev =highest occupied state, Ef = Fermi energy,Eg = energy gap.

In 1961, Shockley and Queisser published apaper on the limiting efficiency of a single p-njunction solar cell [2]. They formulated a de-tailed balance theory that modeled the limit-ing efficiency of an ideal cell. Basically, de-tailed balance is the balance between photonsabsorbed by a solar cell and photons emittedby a solar cell. This will be explained in moredetail in a later section. Efficiency limits forsingle-junction cells were found to be 30% forband gaps of 1.1 eV under a 6000K blackbodyspectrum and a cell temperature of 300K.

The usefulness of the detailed balance the-ory has become apparent to the photovoltaiccommunity [3]. Not only can the theoryprovide limiting efficiencies to single-junctioncells, but it can be extended to calculate thelimiting efficiencies using different spectra (byusing a spectrum’s numerical data) and the-oretical efficiencies of muti-junction or tan-dem cells. Tandem cells consist of severalsingle-junction solar cells of decreasing bandgaps stacked on top of each other. Junctionsare stacked such that their band gap ener-gies are decreasing from top to bottom, i.e.,Eg1 > Eg2 > Eg3 as shown in Fig.2. Stacking

junctions in this manner provides an ”energyfilter” for photons. Photons of high energyare absorbed by the top junction, mid energyphotons pass through the top junction andare absorbed by the middle junction and lowenergy photons pass through the top and mid-dle junctions and are absorbed by the bottomjunction. In order to stack cells on top of one

Figure 2: Latticed matched triple-junctioncell. InGap-1.85eV, GaAs-1.42 eV, Ge-0.66eV.

another, one could simply glue each cell to-gether with conducting cement. However, thisis not mechanically sound. To fix this, cellsare epitaxially and monolithically grown, oneon top of the other. Which materials are usedfor each junction is constrained by the materi-als’ lattice constant. Each material must havethe same lattice constant, otherwise strain isintroduced to the system. Strain causes de-fects where the two materials meet which re-sults in a decrease in efficiency [4]. The stackshown in Fig.2 is an example of a lattice-matched tandem solar cell. DeVos, throughthe use of detailed balance, calculated the the-oretical maximum efficiency of a tandem cellwith three junctions to be 49% under a 6000Kblack body spectrum with energy gap valuesof: Eg1=2.3 eV, Eg2=1.4 eV, Eg3=0.8 eV [5].In addition to modeling triple-junction solarcells the detailed balance theory can be fur-ther extended to model any number of junc-tions. C.H. Henry used graphical techniquesin conjunction with detailed balance to de-termine the theoretical efficiency of a tandem

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cell with 36 cells to be 72% [6]. Others havegeneralized the theory to calculate the effi-ciency of an infinite number of cells. Thisefficiency has been reported to be 86.8% atfull concentration under a 6000K black bodyspectrum [3]. From a fabrication perspective,it is more practical to model triple-junctioncells. This paper explores different methodsby which triple-junction cells are modeled andscrutinizes the affects of concentration andtemperature.

The Thermodynamic Limit

Before the detailed balance theory is pre-sented, it is important to know the maximumefficiency of a photoconverter based only onthermodynamics. Here, only the sun and aphoto converter are considered in a closed sys-tem as detailed in Fig.3 [7].

Figure 3: Model for Carnot Efficiency of pho-toconverter.

In this system, consider the energy flux inci-dent from the sun as well the entropy flux.The sun’s energy flux is incident on a photoconverter which has output flux of work, heatand entropy. The photo conversion processthat takes place inside the photo converter isassumed to cause some amount of entropy in-crease. The first two laws of thermodynamicsare utilized here.

Es = W + Q (1)

Ss + SG = Q/TA (2)

Es is defined as the energy flux from the sun,Ss is the resulting entropy flux of the sun de-fined as Es/Ts. W is the output work flux ofthe converter, Q is the output heat flux of theconverter and TA is the ambient (surround-ing) temperature, which in this case is 300K.SG is the rate of change of the internal en-tropy of the converter, the entropy that is dueto its internal processes such as the transmis-sion, absorption or conversion of sunlight. Bycombining equations 1 & 2, a conversion effi-ciency can be obtained by dividing the outputwork flux of the photo converter by the inputenergy flux of the sun.

η = W/Es = (1− TA/Ts)− TASG/Es (3)

This model gives a maximum efficiency of95% by assuming that SG = 0. This ap-proximation gives the Carnot Efficiency of aphoto converter. However, this is not phys-ically valid because energy transfer betweentwo black bodies of different temperature re-sults in unavoidable entropy production [8].Shockley and Queissar explain why the ther-modynamic limit is not sufficient in providinga fundamental limit by comparing a solar cellto a steam power plant:

”The situation at present may be understoodby analogy with a steam power plant. Ifthe second law of thermodynamics were un-known, there might still exist quite good cal-culations of the efficiency of any given con-figuration based on heats of combustion, etc.However, a serious gap would still exist sinceit would be impossible to say how much theefficiency might be improved by reduction ofbearing friction, improving heat exchangers,etc. The second law of thermodynamics pro-vides an upper limit in terms of more funda-mental quantities such as the temperature ofthe exothermic reaction and the temperatureof the heat sink. The merit of a given powerplant can then be apprised in terms of thelimit set by the second law.”

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A more reasonable fundamental upper limitcan be obtained by considering the fundamen-tal loss mechanisms of a solar cell. These lossmechanisms can be attributed to three funda-mental factors. Photons of energy hν > Egwill excite electrons with an energy higherthan the band edge in which they collidewith lattice atoms, thus loosing energy, andquickly (on the order of 10−15 seconds) relaxto the band edge. This is known as thermal-ization (process 1 in Fig.1). Photons of en-ergy hν < Eg will pass through the material,not contributing to electron-hole pair genera-tion. This is referred to as transmission loss.The amount of efficiency lost through ther-malization and transmission will be shown ina later section. The third fundamental lossmechanism is due to radiative recombination,in which an excited electron will recombinewith a hole in the valance band resulting in anemitted photon (process 2 in Fig.1). The lossdue to spontaneous radiation is unavoidablewhich means that some portion of the powerfrom the sun can never be fully utilized [2].

Detailed Balance

The method of detailed balance is the balancebetween the generation and recombination ofelectron-hole pairs due to the absorption ofphotons from the sun and spontaneous recom-bination, respectively. The first assumptionthat is made in the detailed balance modelis that the photo converter is an absorber ofcharacteristic energy gap (Eg). In regards toelectron-hole generation, consideration mustbe placed on the number of photons that areincident from the sun. Photons incident onthe photo converter will create a current (Jsc)that is a result of electron-hole pair generationand is defined as:

Jsc = qX

∫ ∞0

(1−R(E))α(E)N(E) dE (4)

Where q is the elementary charge, X is con-centration, R(E) is the reflection coefficient,α(E) is the absorption coefficient, and N(E)is the number of photons incident per unittime per area [9]. The N(E) quantity couldbe modeled many different ways. For pur-poses of simulating solar efficiency, N(E) canbe chosen to be a 6000K blackbody, or ASTM(American Society for Testing and Materials)numerical data of extra-terrestrial (AM0) orterrestrial (AM1.5) spectra as shown in Fig.4,taken from Green [10].

Figure 4: Comparison of 6000K BB, AM0 andAM1.5 spectra from Ref. [10].

Modeling the incoming photon flux as a6000K blackbody gives N(E) as:

N(E) =2Fsh3c2

(E2

eE/kBTs − 1

)(5)

Where

Fs = πsin2(θsun) (6)

This is a geometrical factor that arises fromintegrating the photon flux density of the sunover the relevant angular range. This factoraccounts for the decreasing angular range overwhich the sun acts as distance from the sun isincreased. θsun is the half angle subtended bythe radiating body to where the flux is mea-

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sured. In the case of the sun seen from theearth, θsun = 0.26 [9].

In order for this model to provide an upperlimit to efficiency, a few simplifying assump-tions must be made. The material that ismodeled is non-reflecting, which means thatof all the photons that are incident on the ma-terial, none are reflected. In this case, R(E) =0. Another assumption that is made is thatthe absorption coefficient is a step functionsuch that

α(E ≥ Eg) = 1 (7)α(E < Eg) = 0 (8)

In other words, all photons of energy greaterthan the band gap energy are absorbed andall photons of energy less than that are notabsorbed. Considering these assumptions,Eqn.4 can now be written as:

Jsc = qX

∫ ∞Eg

N(E) dE (9)

In addition to photons received from the sun,the solar cell will also receive thermal photonsfrom its surroundings. The ambient is also as-sumed to act as a black body at temperatureTa. This contribution is often negligible, butat high temperatures, low concentrations, orvery low band gap (Eg < 0.3eV ), it is not. Itis expressed as:

Jamb =2πqh3c2

(1−XFs)∫ ∞Eg

E2

eE/kBTa − 1dE

(10)The (1-XFs) term accounts for the angularrange of which the thermal photons act. Ther-mal photons will act on a cell over the remain-ing range that the solar photons do not act.This is illustrated in Fig.5. Considering thesolar cell itself, under illumination and/or ex-ternal bias, some of the electrons have raisedelectrochemical potential energy in which theentire system develops a chemical potential∆µ = qVm , where is the chemical potentialand V is the applied voltage [7]. Under suchconditions, spontaneous emission is increased;

Figure 5: Angular range of thermal photons.

the rate of which depends on the chemical po-tential. This is very similar to the workings ofa Light Emitting Diode (LED). The photonsthat are emitted as a result of recombinationare assumed to radiate like a blackbody at thematerial’s temperature, Ta (here it is assumedthat the cell and the ambient are in thermody-namic equilibrium). Using a generalized formof Plank’s radiation law, the radiative currentcan be written as the following [9].

Jrad =2πqh3c2

∫ ∞Eg

E2

e(E−∆µ)/kBTa − 1dE (11)

Finally, the net current is obtained such that,

J = Jsun + Jamb − Jrad (12)

and the efficiency is,

η =V J(V )Ps

(13)

Where Ps is the input power from the sun andVJ(V) is the power of the cell at any voltage.As the chemical potential increases, the cur-rent in Eqn.12 decreases while the power in-creases until a maximum is reached. Furtherincreases of the chemical potential result indecreasing power until it is reduced to zero.This will be illustrated in the next section.

Simulation Methods

The detailed balance model described abovewas utilized to model single and tandem solar

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cells using Matlab. The code for the singlejunction model uses Eqns. 4, 10, 11, 12, and13 for a 6000K blackbody spectrum. The taskfor the code is to calculate the net currentof a single junction cell over a range of volt-ages for a given Ts, Ta, Eg, and X. The volt-age is varied from zero to Eg/q. For a blackbody spectrum, the current integrals are cal-culated using Gauss-Kronrod quadrature. Ifthe spectrum being used is one that is purelynumerical data, such as an AM0 or AM1.5spectrum, the Jsc integral is solved using atrapezoidal rule. Incident power from the sunalso changes as the spectrum is changed. Cod-ing for the single junction model can be foundin Appendix C. The result of a single junc-tion simulation is shown in Fig.6. The maxi-

Figure 6: J-V curve for Silicon showing maxpower point and other parameters.

mum power point is the point at which outputpower is maximized; graphically this can eas-ily be seen to occur at the ”knee” of the J-Vcurve (see Fig.6). Corresponding to it are itsmaximum operating current, Jm and maxi-mum operating voltage Vm. The area underthe dashed square in Fig.6 directly determinesthe maximum efficiency of the cell such that

ηmax =VmJmPs

(14)

Simulations such as this were done for a rangeof band gaps. The maximum efficiencies forband gaps ranging from 0.5 eV to 2.5 eV un-der a 6000K black body spectrum are shown

in Fig.7. This was done by modifying thesingle-junction code to deal with a range ofband gap values and store the efficiency foreach band gap. It is clear that maximum ef-ficiency depends on which type of materialis chosen for the solar cell. An efficiency of31% was obtained for a band gap 1.3 eV anda 6000K black body input spectrum, whichagrees well with previous work [2] [3] [11].Since the solar spectrum has such a broad

Figure 7: Dependence of efficiency on BandGap energy for 6000K blackbody spectrum.

range of wavelengths, only a portion of thespectrum can be utilized by a single junctionsolar cell. Using the single-junction simula-tion, the amount of the solar spectrum thatis actually absorbed and lost was also calcu-lated. Fig.8 shows an AM0 spectrum cour-tesy of the ASTM. The portion shaded inred corresponds to the amount of the solarspectrum that is actually usable by a single-junction cell, in this case GaAs. The lossesdue to thermalization and transmission arealso illustrated and were found to be 33% and23% respectively. These were found by nu-merically integrating the areas not shaded inred and finding their ”would be” efficiency.An initial program was then written to simu-late tandem or triple junction cells based onthe above programs. In order to simulate thetriple-junction cell, each junction was simu-lated separately. The assumptions were made

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Figure 8: Amount of Solar Spectrum (AMO)usable by single-junction solar cell (GaAa).

that since the junctions are stacked on top ofone another in series, the cells could be mod-eled as an equivalent series circuit (Fig.9) suchthat the output current would be limited bythe junction producing the least amount ofcurrent. In this case the assumption was also

Figure 9: Equivalent series circuit of triple-junction solar cell. 1 is the top junction, 2 isthe middle junction, 3 is the bottom junction.

made that the voltages of each junction addand that each junction will operate at its max-imum power point. Using these assumptions,the output power of the triple-junction cell is

given as:

Pcell = Jmin

3∑n=1

Vn (15)

Jmin = min(J1, J2, J3) (16)

Treating each cell separately in this mannerspeeds calculation time, but results in moreerror. It will be referred to as the ”I-Vmethod”. An example of a simulation donewith the I-V method is shown in Figs. 10and 11. Here a State-of-the-art (SOA) lat-tice matched InGaP/GaAs/Ge cell was cho-sen to be modeled. Fig.10 shows the J-Vcurves of InGaP (Eg = 1.85eV), GaAs (Eg= 1.42eV), and Ge (Eg = 0.66eV). Thesecurves were modeled using detailed balanceas above, but using the following integrationranges. Top Junction (InGaP): 1.85eV to ∞.Middle Junction (GaAs): 1.42eV to 1.85eV.Bottom junction (Ge): 0.66eV to 1.42eV.Fig.11 shows an AM0 spectrum with the cor-responding modeled SOA absorption. Clearly,by splitting the spectrum both thermalizationand transmission loss are reduced. This is themain advantage of multi-junction solar cells.

Figure 10: I-V curves of each junction in aState-of-the-Art tandem cell. Real operatingpoints are circled.

While convenient, the I-V method is less ac-curate because the assumption was made thatthe operating voltages of each junction in thetandem are the same as if the junctions wereindependent of each other. Under real oper-ation,the limiting current of the tandem cell

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Figure 11: Absorption of triple-junction cell.Compared to single-junction, more of thespectrum can be utilized. Thermaliztion andtransmission efficiency loses are reduced.

forces the junctions with larger currents to op-erate with lower currents and at higher volt-ages than normal. This is shown in Fig.10.The points on each J-V curve that the mini-mum current line (green) intersects are takento be the operating points of the tandem cell.

In principle, one could make Fig.10 for all pos-sible band gap configurations, find the mini-mum current, find which sub-cell is currentlimiting and then calculate the other corre-sponding sub-cell voltages. However, this isnot practical and results in time-consumingand processor intensive code. The true op-erating points can be found directly throughthe use of Lagrange Multipliers [11]. This is amethod by which a function can be maximizedunder given constraints. In the case of thetandem cell, Eqn.15 would need to be maxi-mized under the constraint that each junctionmust have the same current. That is

J1 − J2 = 0 (17)J2 − J3 = 0 (18)

Given these constraints, Eqn.15 can be rewrit-

ten as a function voltage:

F (V1, V2, V3) =3∑

n=1

JnVn + λ12(J1 − J2) + λ23(J2 − J3)

(19)Here the Lagrange multipliers λ12 and λ23

have been introduced. To simplify the math-ematics, dimensionless parameters were usedto scale Eq.19. They are

β =qVnkTa

(20)

α =E

kTa(21)

Now, Eq.19 can be written as

F (β1, β2, β3) =∑

jnβn + λ12φ1 + λ23φ2 (22)

Where

jn = XFs

∫α2dα

eαTa/Ts − 1+ (1−XFs)

∫α2dα

eα − 1−

∫α2dα

eα−βn − 1(23)

jn =Jnh

3c2

2πk3T 3a

(24)

φ1 = j1 − j2 (25)φ2 = j2 − j3 (26)

In order to maximize Eqn.22 and hence thepower of the cell, partial derivatives must betaken with respect to β1, β2, β3 and set equalto zero. Explicitly,

∂F

∂β1= 0 (27)

∂F

∂β2= 0 (28)

∂F

∂β3= 0 (29)

The explicit derivatives can be found in Ap-pendix B. With equations 25-29, there wouldbe 5 equations and 5 unknowns, the threescaled potentials and the two Lagrange multi-pliers. First the Lagrange multipliers are de-termined analytically which saves on compu-tation time. After doing so (see Appendix A), the final result was found to be:∑

(βn +jnj′n

) = 0 (30)

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Where

j′n =djndβn

= −14

∫α2csch(

α− βn2

) dα (31)

Now, equations 25, 26 and 30 can be simul-taneously solved numerically to yield eachjunction’s operating voltage and the resultingmatched current.

To code this, a separate file was made that in-cluded the three equations to be solved. Thisfile would be called on from the main pro-gram. In the main program, starting pointsfor solving the three equations had to be cho-sen. Values close to Vm1, Vm2, Vm3, were cal-culated and scaled to be the initial startingpoints for solving equations 25, 26 and 30.These were calculated using an approximationmethod formulated by C.H. Henry [6]. His re-sult for approximating Vm was

qVm = qVoc − kT ln(1 +qVmkT

) (32)

Once the initial starting points were found,the short-circuit current for each junction wascalculated. Equations 25, 26 and 30 were thensolved using a built-in solver in Matlab calledfsolve. The coding of this method can befound in Appendix C. The maxima found bythis method are local maxima of a user de-fined set of band gap energies. Maximum ef-ficiency and optimum band gaps for a specificspectrum, temperature and concentration areof great importance. To find the global maxi-mum, Eqn.22 can be rewritten to include de-pendencies of α1, α2, and α3 such that:

F (α1, α2, α3, β1, β2, β3) =∑

jnβn + λ12φ1 + λ23φ2

(33)By taking three more partial derivatives suchthat:

∂F

∂α1= 0 (34)

∂F

∂α2= 0 (35)

∂F

∂α3= 0 (36)

Using these derivatives in conjunction withequation 30 and the constraints on the cur-rents, there are 6 equations with 6 unknowns.Solving these numerically yields the optimumband gaps, operating points, and maximumefficiency of a tandem cell for a specific spec-trum, temperature and concentration.

Results

Using the I-V method the bottom junction ofthe triple junction cell was fixed at the Geband gap 0.66 eV while the top and middlejunctions were allowed to vary. The top andmiddle junctions vary such that the condition:Eg1 > Eg2 > Eg3, is still satisfied. In thetandem model, in order to obtain the absolutemaximum efficiency theoretically possible, theconstraint that each junction must be lattice-matched was ignored. Therefore, maximumefficiency for every possible energy gap config-uration was found. The results of which areillustrated in the iso-efficiency plot in Fig.12.The same thing was done for the local maxima

Figure 12: Contour plot of efficiency for triplejunction solar cell with fixed Ge bottom junc-tion for 6000K blackbody spectrum. The topcontour line is 45% and they decrease in stepsof 5 down to 5%. Maximum efficiency is 47%.

Lagrange method. This is shown in Fig.13:The main interests are the top and middleband gaps that provide the efficiency at the

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Figure 13: Lagrange Iso-Efficiency plot for6000K Black Body.

very peak of the contour plot. This processcan be repeated for the AM0 and AM1.5Gspectra. The results of which are found inTable 1 located in Appendix A.

The results agree well with literature result[3] [5] [11]. It can be seen in Table 1 thatthere is a very small discrepancy between theresults of the two models for the maximum ef-ficiency points and corresponding band gaps.However the same is not true once the bandgap values stray away from optimum. Fig.14shows the relative error between the I-V andLagrange methods. The values from the La-grange method are assumed to be the ac-cepted values. The Lagrange contour plot isoverlaid with the error plot to demonstratewhere the least amount of error is. It isclear that the least amount of error is seenat the very peak of the Lagrange contour plot(< 1%). This error arrises because the IVmethod takes the operating current to be thatof the max power point of the limiting cell.However, this current is not the current formax power of the tripe junction cell. In atriple junction cell, none of the cells are oper-ating at their individual max power currentswhen the triple junction cell is at its maxpower. The Lagrange method determines theactual max power current and max voltages ofthe triple junction cell. This statement doesnot apply to simply the global max power, but

at each max power point. Near the global maxeffieciency point, however, all of the cells havenearly the same max power current, and theyare operating at this common current. There-fore, it is clear that the error between the twomethods is small at the global max efficiencypoint. As the energy gap values stray fromthis point, the error increases. The reason forthis is due to the variance in operating volt-ages between the two models. As this differ-ence increases, so does the error. For example,the SOA triple junction cell that was modeledusing the I-V method has an error of approxi-mately 5%, 31% efficiency was obtained usingthe I-V method and 33% was obtained usingthe Lagrange method. It is clear from thisthat for local maxima, the Lagrange methodis a more accurate model. But for the max-imum point, either model yields accurate re-sults. The global maxima Lagrange model is

Figure 14: Relative error between I-V and La-grange methods.

a function of spectrum, temperature and con-centration. This makes it ideal for exploringthe affects of concentration and temperatureon the maximum efficiency point of a tandemcell. In order to do this, the bottom cell wasfixed to Ge just as before. As the Ge sub-cellwas fixed, the band gap was allowed to varywith temperature based on the empirical re-lation [12]

Eg = 0.742− 4.8x10−4T 2

(T + 235)(37)

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The temperature used in the model was thenvaried from 80-400K. This was done for con-centrations of 1 - 1000 suns. Current concen-tration system operate at 500X but are ex-pected to begin to move toward 1000X. As

Figure 15: Effects of temperature and con-centration on Efficiency for tandem cell under6000K Black Body.

Figure 16: Effects of temperature and concen-tration on optimum top cell energy gap under6000K black body.

expected the maximum efficiency varies loga-rithmically with increasing concentration andthe efficiency varies linearly with temperature. It can be seen that in both Fig.16 and 17, theband gap energy curves up slightly at highertemperatures. This curving phenomenon isdue to the added contribution from the ambi-ent current (Eqn.10) at higher temperatures.This is also evident because as concentrationincreases, positive concavity seen at higher

Figure 17: Effects of temperature on optimummiddle junction band gap energy under 6000Kblack body.

temperatures becomes more and more shal-low. This is demonstrated in Eqn.10 by thefact that the ambient current becomes less sig-nificant with increasing concentration. Theboxed-in sections of Figs. 15, 16 and 17 high-light practical applications where this infor-mation might be useful. Satellites sent toouter planets experience Low Intensity andLow Temperature (LILT). Knowing how ef-ficiency is affected by such low temperaturesand concentrations would be the determiningfactor in powering the satellite. At one sunconcentration, efficiencies in this region rangefrom 63% to 56% for temperatures rangingfrom 80K to 190K respectively. The samecan be said for satellites put into Low EarthOrbit (LEO). Satellites in LEO are exposedto temperatures between 190 and 350K every90min. These efficiencies range from 56% to45% at one sun concentration. So, knowinghow temperature affects the solar cells thatpower the satellite is useful. Concentratorsystems and satellites sent to the inner planets(Mercury, Venus) also operate at high tem-peratures. Models such as this could help de-signers determine what materials to use andhow the system will behave at various tem-peratures.

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Conclusion

Detailed balance models for single-junctionand tandem cells were made using Matlab.Local and global models using Lagrange mul-tipliers were made and compared to previousresults. Maximum efficiency points of the I-Vmodel boast less than 1% error when com-pared to that of the local Lagrange model.The global Lagrange model was used to ex-plore the effects of concentration and tem-perature in a more accurate fashion than wasdone previously. The results of the Lagrangemodels can be useful in deciding which mate-rials to use for a tandem solar cell that wouldproduce a maximum in efficiency. In addi-tion, exploring the effects of temperature andconcentration demonstrates, in general, whichband gap energies would produce a max effi-ciency for a given concentration and tempera-ture. This especially important for the designof solar cells that are used in a variety of dif-ferent environments where the temperature isconstantly changing, such as a satellite orbit-ing earth.

References

[1] J. F. Geisz, D. J. Friedman, J. S. Ward,A. Duda, W. J. Olavarria, T. E. Mo-riarty, J. T. Kiehl, M. J. Romero, A.G. Norman, and K. M. Jones: 40.8%Efficient inverted triple-junction solarcell with two independently metamor-phic junctions, Applied Physics Letters,Vol 93, (2008)

[2] W. Shockley, H. J. Queissar: DetailedBalance Limit of Efficiency of p-n Junc-tion Solar Cells, Journal of AppliedPhysics, Vol 32, pp. 510-519 (1961)

[3] G. L. Araujo, A. Marti: Absolute lim-iting efficiencies of photovoltaic energyconversion, Solar Energy Materials andSolar Cells, Vol 33, pp. 213-244 (1994)

[4] A. W. Bett et al : Highest EfficiencyMulti-Junction Solar Cell for Terrestrialand Space Applications, Hamburg, Ger-many: WIP, 24th European Solar En-ergy Conference and Exhibition, pp. 1-6.(2009)

[5] A. De Vos: Detailed Balance Limit ofthe efficiency of tandem cells, J. Phys,D: Appl. Phys., Vol 13, (1980)

[6] C. H. Henry: Limiting Efficiencies ofideal single and multiple energy gap ter-restrial solar cells, J. Appl. Phys., Vol 51,pp. 4494-4499. (1980)

[7] M. A. Green: Third Generation Photo-voltaics:Advanced Solar Energy Conver-sion, Springer, (2003)

[8] A. H. Pauwels, A. De Vos: Determina-tion of the maximum efficiency solar cellstructure, Solid State Electronics, Vol 24

[9] J. Nelson: The Physics of Solar Cells,Imperial College Press, (2003)

[10] M. A. Green: Solar Cells: OperatingPrinciples, Technology, and System Ap-plications, Prentice-Hall, Inc, (1982)

[11] R. Aguinaldo: Modeling Solutions andSimulations for Advanced III-V Photo-voltaics Based on Nanostructures, (2008)

[12] Physical Properties of Ge Semicon-ductors on NSM, http://www.ioffe.ru/SVA/NSM/Semicond/Ge/bandstr.htmlTemperature

[13] A. W. Bett, et al : Development of III-V-Based Concentrator cells and Their Ap-plication in PV-Modules, New Orleans,29th IEEE PVSC, (2002)

[14] A. S. Brown, M. A. Green: LimitingEfficiency for Current-Constrained TwoTerminal Tandem Cell Stacks, Prog.Photovolt: Res. Appl., Vol 10, pp. 299-307, (2002)

12

Appendices

A Max efficiencies for different spectra

Spectrum LagrangeMax. Eff.(%)

IV Max.Eff. (%)

LagrangeEg1 (eV)

IV Eg1 (eV) LagrangeEg2 (eV)

IV Eg2 (eV)

6000K BB 48.10 48.0 1.86 1.86 1.18 1.19AM0 47.48 47.4 1.75 1.77 1.11 1.12

AM1.5G 50.58 50.3 1.79 1.79 1.19 1.2

B Lagrange Multipliers and Derivatives

Given the function that is to be maximuize:

F (α1, α2, α3, β1, β2, β3) =∑

jnβn + λ12φ1 + λ23φ2

Take derivatives w.r.t. the 3 β parameters:

∂F

∂β1= j′1β1 + j1 + λ12j

′1 = 0

∂F

∂β2= j′2β2 + j2 − λ12j

′2 + λ23j

′2 = 0

∂F

∂β3= j′3β3 + j3 + λ23j

′3 = 0

The Lagrange Multipliers can now be solved for using substitution or elimination:

λ12 =j′3j2 + j′2j3 + j′2j

′3(β2 + β3)

j′2j′3

λ23 =−(j1j′2 + j2j

′1 + j′1j

′2(β1 + β2))

j′1j′2

This gives the result: ∑(βn +

jnj′n

) = 0

The Lagrange Multipliers can now be written as:

λ12 = −β1 −j1j′1

13

λ23 = −β3 −j3j′3

Now the α derivatives can be done. It is necessary to substitute the Lagrange Multipliers wherenecessary which gives:

∂F

∂α1= j1

∂j1/∂α1

∂j1/∂β1+ j2

∂j2/∂α1

∂j2/∂β2= 0

∂F

∂α2= j2

∂j2/∂α2

∂j2/∂β2+ j3

∂j3/∂α2

∂j3/∂β3= 0

∂F

∂α3= 0

In order to evaluate the α derivatives, Lebnitz rule is necessary.

C Sample Code

Below are a few programs that I wrote over the course of Capstone I and II. Many others werewritten, but were primarily based off of the programs presented here.

%%%%%%%%%%%%%%%%%%%%%% Sing le−Junct ion Model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% This model was made as a func t i on in order to be used with other programs% I t r e tu rn s the opera t ing po int o f an absorber in the energy range o f Eg to% some upper l i m i t . I t can e a s i l y be and was used to f i n d e f f i c i e n c y o f a s i n g l e% junc t i on c e l l , though t h i s v e r s i on i s not des igned to . The spectrum% (0 f o r BB, 1 f o r AM0, 2 f o r AM1. 5G, 3 f o r AM1. 5D) , concent ra t i on (X)%and ambient temperature (Ta) can be s p e c i f i e d .

f unc t i on [Vm, Jm] = dbsj (Eg , spectrum , UpLim ,X, Ta)

%Set Constantsq=1.6021765e−19; h=6.626069e−34; c =299792458; k=1.38065e−23; f =2.16e−5;Ts=6000;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Set vo l tage to an appropr ia t e rangeV=l i n s p a c e (0 ,Eg , 4 0 0 ) ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Now, we w i l l s e l e c t the chosen spectrum and c a l c u l a t e the cor re spond ing shor t%c i r c u i t cur rent

i f spectrum==0;

14

SunFlux = @(E) 2∗ pi /(hˆ3∗ c ˆ2) ∗ E.ˆ2 . / ( exp (E/( k∗Ts))−1) ;%the 1e−1 f a c t o r conver t s A/mˆ2 to mA/cmˆ2Jsc = q ∗ X ∗ f ∗ quadgk ( SunFlux , Eg∗q , UpLim∗q )∗1 e−1;%Calcu la te i n c i d e n t energySunEnergy = @(E) 2∗ pi /(hˆ3∗ c ˆ2) ∗ E.ˆ3 . / ( exp (E/( k∗Ts))−1) ;Ps = X ∗ f ∗ quadgk ( SunEnergy ,0 , 22∗ q )∗1 e−1;

e l s eload spectrumFluxi f spectrum==1;

SunFlux=AM0; lambda=lam0 ; Ps=X∗13 6 . 61 ;e l s e i f spectrum==2;

SunFlux=AM1p5G; lambda=lam1p5 ; Ps=X∗10 0 . 04 ;e l s e i f spectrum==3;

SunFlux=AM1p5D; lambda=lam1p5 ; Ps=X∗90 . 014 ;e l s e

d i sp ( ’ S e l e c t i o n i s not a p p l i c a b l e f o r So la r Spectrum ’ ) ;c l e a r AM0 AM1p5G AM1p5D lam0 lam1p5 ;re turn

endc l e a r AM0 AM1p5G AM1p5D lam0 lam1p5 ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%lambdaG1 and lambdaG2 i s wavelength that corresponds to the energy gap%and upper l i m i t r e s p e c t i v e l ylambdaG = h∗c /(Eg∗q ) ∗ 1e9 ;lambdaG2 = h∗c /(UpLim∗q )∗1 e9 ;minPt = f i n d ( lambda>=lambdaG2 , 1 , ’ f i r s t ’ ) ;maxPt = f i n d ( lambda<=lambdaG , 1 , ’ l a s t ’ ) ;Jsc = q ∗ X ∗ t rapz ( lambda ( minPt : maxPt ) , SunFlux ( minPt : maxPt ))∗1 e−1;

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Here , we w i l l c a l c u l a t e the cur rent from ambient f lux , cur rent l o s t from%r a d i a t i v e recombination , and the t o t a l r e s u l t i n g cur rentAmbientFlux = @(E) 2∗ pi /(hˆ3∗ c ˆ2) ∗ E.ˆ2 . / ( exp (E/( k∗Ta))−1) ;Jamb = q∗(1−(X∗ f ) ) ∗ quadgk ( AmbientFlux , Eg∗q , UpLim∗q ) ;Jrad=ze ro s (1 , l ength (V) ) ; Jdark=ze ro s (1 , l ength (V) ) ; J=ze ro s (1 , l ength (V) ) ; z =1;whi l e z<l ength (V) ;

Ce l lF lux = @(E) 2∗ pi /(hˆ3∗ c ˆ2) ∗ E.ˆ2 . / ( exp ( (E−(q∗V( z ) ) ) / ( k∗Ta))−1) ;Jrad ( z)= q ∗ quadgk ( Cel lFlux , Eg∗q , UpLim∗q ) ;Jdark ( z ) = (Jamb − Jrad ( z ) )∗1 e−1;J ( z ) = ( Jsc + Jdark ( z ) ) ;i f J ( z )<0;

J ( z +1: l ength ( J ) ) = NaN;z=length (V) ;

e l s e

15

z=z +1;end

end% Find maximum power , maxpower po int and corre spond ing opera t ing vo l tage% and cur rent

Power = V.∗ J ;MaxPt = f i n d ( Power==max( Power ) ) ;Vm = V(1 ,MaxPt ) ;Jm = J (1 ,MaxPt ) ;

%Plot Resu l t s%p lo t (V, J ) ; a x i s ( [ 0 Eg 0 2ˆnextpow2 ( J ( 1 ) ) ] )

%%%%%%%%%%%%%%%%%%%%% Trip le−Junct ion I−V Model %%%%%%%%%%%%%%%%%%%%%%%%%%

%Set cons tant s and s e t range o f top two j u n c t i o n sq=1.6021765e−19; h=6.626069e−34; c =299792458; k=1.38065e−23; f =2.16e−5;Ts=6000; Ta=300; spectrum =0; X=1; Eg1=l i n s p a c e ( 1 . 5 0 1 , 2 , 5 0 ) ;Eg2=l i n s p a c e ( 1 , 1 . 5 , 5 0 ) ;

Eg3 = 0.742 − ( 4 . 8 e−4∗(Ta . ˆ 2 ) ) . / (Ta+(235)) ;

% With chosen spectrum , c a l c u l a t e i n c i d e n t energyi f spectrum==0;

SunEnergy = @(E) 2∗ pi /(hˆ3∗ c ˆ2) ∗ E.ˆ3 . / ( exp (E/( k∗Ts))−1) ;Ps = X ∗ f ∗ quadgk ( SunEnergy ,0 , 22∗ q )∗1 e−1;%the 1e−1 f a c t o r conver t s A/mˆ2 to mA/cmˆ2

e l s ei f spectrum==1;

Ps=X∗13 6 . 61 ;e l s e i f spectrum==2;

Ps=X∗10 0 . 04 ;e l s e i f spectrum==3;

Ps=X∗90 . 014 ;e l s e

d i sp ( ’ S e l e c t i o n i s not a p p l i c a b l e f o r So la r Spectrum ’ ) ;r e turn

endend

% I n i t i a l i z e m a t r i c i e sVm1=ze ro s (1 , l ength (Eg1 ) ) ;Vm2=ze ro s (1 , l ength (Eg1 ) ) ;Vm3=ze ro s (1 , l ength (Eg1 ) ) ; Jm1=ze ro s (1 , l ength (Eg1 ) ) ;Jm2=ze ro s (1 , l ength (Eg1 ) ) ; Jm3=ze ro s (1 , l ength (Eg1 ) ) ;V tota l=ze ro s (1 , l ength (Eg1 ) ) ; Jm=ze ro s (1 , l ength (Eg1 ) ) ;

16

MinPt=ze ro s (1 , l ength (Eg1 ) ) ; Jm min=ze ro s (1 , l ength (Eg1 ) ) ;E f f=ze ro s (1 , l ength (Eg1 ) ) ; Jm min12=ze ro s (1 , l ength (Eg1 ) ) ;Jm min23=ze ro s (1 , l ength (Eg1 ) ) ;

% Use the s i n g l e−j unc t i on model to f i n d max power po in t s o f every% combination o f band gaps

f o r i =1: l ength (Eg1)f o r j =1: l ength (Eg1 )

[Vm1( i , j ) ,Jm1( i , j )]= dbs j (Eg1 ( i ) , spectrum , 2 2 ,X, Ta ) ;[Vm2( i , j ) ,Jm2( i , j )]= dbs j (Eg2 ( j ) , spectrum , Eg1( i ) ,X, Ta ) ;[Vm3( i , j ) ,Jm3( i , j )]= dbs j (Eg3 , spectrum , Eg2( j ) ,X, Ta ) ;

% Ensure that cur rent i s always p o s i t i v ei f Jm1( i , j )<0

Jm1( i , j )=0;e l s e i f Jm2( i , j )<0

Jm2( i , j )=0;e l s e i f Jm3( i , j )<0;

Jm3( i , j )=0;end

endend

% Find minimum currentJm min12 ( i , j )=min (Jm1( i , j ) ,Jm2( i , j ) ) ;Jm min23 ( i , j )=min (Jm2( i , j ) ,Jm3( i , j ) ) ;Jm min( i , j )=min ( Jm min12 ( i , j ) , Jm min23 ( i , j ) ) ;

% Ca lcu la te t o t a l vo l t ageV tota l ( i , j )=Vm1( i , j )+Vm2( i , j )+Vm3( i , j ) ;

% Ca lcu la te e f f i c i e n c y f o r each po intEf f ( i , j )=( V tota l ( i , j )∗Jm min( i , j ) )/ Ps ;

endend

% Plot r e s u l t sv = [ 7 5 , 70 , 65 , 60 , 55 , 50 , 45 , 40 , 35 , 30 , 25 , 20 , 15 , 10 , 5 ] ;[ Eg2 , Eg1 ] = meshgrid (Eg2 , Eg1 ) ;colormap ( j e t ) ;[C, h]= contour (Eg2 , Eg1 , ( E f f ∗100) , v ) ;c l a b e l (C, h ) ;t i t l e ( ’6000K Black Body ’ ) ;x l a b e l ( ’ Eg2 (eV ) ’ ) ;y l a b e l ( ’ Eg1 (eV ) ’ ) ;

17

%%%%%%%%%%%%%%%%%%%Lagrange Local Tr ip le−Junct ion Model%%%%%%%%%%%%%%%%%%%%

%constant sq=1.6021765e−19; h=6.626069e−34; c =299792458; k=1.38065e−23; f =2.16e−5;Ts=6000; Ta=300; X=1; spectrum =0;

%Energy GapsEg1 = l i n s p a c e ( 1 . 5 0 1 , 2 , 1 0 0 ) ; Eg2 = l i n s p a c e ( 1 , 1 . 5 , 1 0 0 ) ; Eg3 = 0 . 6 6 ;

%Find Max Voltages f o r s t a r t i n g po in t sVm1=ze ro s (1 , l ength (Eg1 ) ) ; Voc1=Vm1; Vm2=Vm1; Voc2=Vm1;f o r i =1: l ength (Eg1)[Vm1( i ) , Voc1 ( i ) ] = henrydb (Eg1( i ) , 22 ,Ta , spectrum ) ;[Vm2( i ) , Voc2 ( i ) ] = henrydb (Eg2 ( 1 ) , Eg1 ( i ) ,Ta , spectrum ) ;end[Vm3, Voc3 ] = henrydb (Eg3 , Eg2 ( 1 ) ,Ta , spectrum ) ;

%d imens i on l e s s f a c t o r sA1 = (Eg1∗q )/ ( k∗Ta ) ; A2 = (Eg2 .∗ q ) . / ( k∗Ta ) ; A3 = (Eg3∗q )/ ( k∗Ta ) ;AMax = (22∗q )/ ( k∗Ta ) ;

i f spectrum==0;%c a l c u l a t e d imens i on l e s s Jsc f o r each junc t i onSunFlux = @(A) A.ˆ2 . / ( exp (A∗(Ta/Ts ))−1) ;j s c 2=ze ro s (1 , l ength (Eg2 ) ) ; j s c 3=ze ro s (1 , l ength (Eg2 ) ) ; j s c 1=j s c 2 ;%Ca lcu la te Inc iden t EnergySunEnergy = @(A) A.ˆ3 . / ( exp (A∗(Ta/Ts ))−1) ;ps = f ∗ X ∗ quadgk ( SunEnergy , 0 ,AMax) ;%Ca lcu la te shor t c i r c u i t cu r r en t sf o r i =1: l ength (Eg1)

f o r j =1: l ength (Eg2 )j s c 1 ( i , j ) = f ∗ X ∗ quadgk ( SunFlux , A1( i ) ,AMax) ;j s c 2 ( i , j ) = f ∗ X ∗ quadgk ( SunFlux , A2( j ) ,A1( i ) ) ;j s c 3 ( i , j ) = f ∗ X ∗ quadgk ( SunFlux , A3 , A2( j ) ) ;

endend

e l s eload spectrumFluxi f spectrum==1;

SunFlux=AM0; lambda=lam0 ; Ps=X∗13 66 . 1 ;ps=(Ps∗hˆ3∗ c ˆ2)/(2∗ pi ∗kˆ4∗Ta ˆ 4 ) ;

e l s e i f spectrum==2;SunFlux=AM1p5G; lambda=lam1p5 ; Ps=X∗10 00 . 4 ;ps=(Ps∗hˆ3∗ c ˆ2)/(2∗ pi ∗kˆ4∗Ta ˆ 4 ) ;

e l s e i f spectrum==3;SunFlux=AM1p5D; lambda=lam1p5 ; Ps=X∗90 0 . 14 ;ps=(Ps∗hˆ3∗ c ˆ2)/(2∗ pi ∗kˆ4∗Ta ˆ 4 ) ;

e l s e

18

di sp ( ’ S e l e c t i o n i s not a p p l i c a b l e f o r So la r Spectrum ’ ) ;c l e a r AM0 AM1p5G AM1p5D lam0 lam1p5 ;re turn

endc l e a r AM0 AM1p5G AM1p5D lam0 lam1p5 ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%lambdaG i s wavelength that corresponds to the energy gaplambdaGMax = h∗c /(22∗q ) ∗ 1e9 ;lambdaG1 = ( ( h∗c ) . / ( Eg1 .∗ q ) ) . ∗ 1 e9 ;lambdaG2 = ( ( h∗c ) . / ( Eg2 .∗ q ) ) . ∗ 1 e9 ;lambdaG3 = ( ( h∗c )/ ( Eg3∗q ))∗1 e9 ;

% Find max po in t s and shor t c i r c u i t cu r r en t sminPtGMax = f i n d ( lambda>=lambdaGMax , 1 , ’ f i r s t ’ ) ;maxPtG1=ze ro s (1 , l ength (Eg1 ) ) ; maxPtG2=maxPtG1 ;Jsc1=ze ro s ( l ength (Eg1 ) , l ength (Eg1 ) ) ;Jsc2=Jsc1 ; Jsc3=Jsc1 ;f o r i =1: l ength (Eg1)

f o r j =1: l ength (Eg1 )maxPtG1( i ) = f i n d ( lambda<=lambdaG1( i ) , 1 , ’ l a s t ’ ) ;maxPtG2( j ) = f i n d ( lambda<=lambdaG2( j ) , 1 , ’ l a s t ’ ) ;maxPtG3 = f i n d ( lambda<=lambdaG3 , 1 , ’ l a s t ’ ) ;Jsc1 ( i , j ) = q ∗ X ∗t rapz ( lambda (minPtGMax : maxPtG1( i ) ) , SunFlux (minPtGMax : maxPtG1( i ) ) ) ;Jsc2 ( i , j ) = q ∗ X ∗t rapz ( lambda (maxPtG1( i )+1:maxPtG2( j ) ) , SunFlux (maxPtG1( i )+1:maxPtG2( j ) ) ) ;Jsc3 ( i , j ) = q ∗ X ∗t rapz ( lambda (maxPtG2( j )+1:maxPtG3) , SunFlux (maxPtG2( j )+1:maxPtG3 ) ) ;

endend% switch to d imens i on l e s s cur rentj s c 1 = ( Jsc1 ∗hˆ3∗ c ˆ2)/(2∗ pi ∗q∗kˆ3∗Ta ˆ 3 ) ;j s c 2 = ( Jsc2 ∗hˆ3∗ c ˆ2)/(2∗ pi ∗q∗kˆ3∗Ta ˆ 3 ) ;j s c 3 = ( Jsc3 ∗hˆ3∗ c ˆ2)/(2∗ pi ∗q∗kˆ3∗Ta ˆ 3 ) ;

end

% I n i t i a l i z e m a t r i c i e sB01=ze ro s (1 , l ength (Eg1 ) ) ; B02=B01 ; B03=B01 ;j 1=ze ro s (1 , l ength (Eg2 ) ) ; j 2=ze ro s (1 , l ength (Eg2 ) ) ;j 3=ze ro s (1 , l ength (Eg2 ) ) ;B=ze ro s (1 , l ength (Eg2 ) ) ; p=ze ro s (1 , l ength (Eg2 ) ) ;E f f=ze ro s (1 , l ength (Eg2 ) ) ; B0=ze ro s (1 , l ength (Eg2 ) ) ; Bup=ze ro s (1 , l ength (Eg2 ) ) ;Blow=ze ro s (1 , l ength (Eg2 ) ) ; BF1=ze ro s (1 , l ength (Eg2 ) ) ; BF2=BF1 ; BF3=BF1 ;

f o r i =1: l ength (Eg2)% Find i n i t i a l s t a r t i n g po in t s based on Voltages found us ing henry% model above

19

B01( i ,1)=( q∗Vm1( i ) ) / ( k∗Ta ) ; B02( i ,1)=( q∗Vm2( i ) ) / ( k∗Ta ) ; B03( i ,1)=( q∗Vm3)/( k∗Ta ) ;f o r j =1: l ength (Eg2 )

%Solve Legrange Optimizat ion equat ions as coded in l egopt .mB0 = [ B01( i , j ) , B02( i , j ) , B03( i , j ) ] ;opt i ons = opt imset ( ’ Display ’ , ’ i t e r ’ , ’ MaxFunEval ’ , 5 0 0 0 , ’ MaxIter ’ , 5 0 0 0 , . . .’ TolFun ’ , 1 e−12 , ’TolX ’ , 1 e−12);F=@(B) l egopt (B, j s c 1 ( i , j ) , j s c 2 ( i , j ) , j s c 3 ( i , j ) ,A1( i ) ,A2( j ) ,A3 ,AMax) ;[B, resnom ] = f s o l v e (F , B0 , opt ions ) ;

t o l = 1e−10;i f resnom > t o l

breakend

% c a l c u l a t e matched d imens i on l e s s cur rentj 1 ( i , j ) = j s c 1 ( i , j ) − quadgk (@(A) A.ˆ2 . / ( exp (A−B(1))−1) ,A1( i ) ,AMax) ;p( i , j ) = j1 ( i , j ) ∗ (B(1 ) + B(2) + B( 3 ) ) ;E f f ( i , j ) = (p( i , j )/ ps )∗100 ;

% Use c a l c u l a t e d roo t s f o r s t a r t i n g po in t s o f next loop runB01( i , j +1) = B( 1 ) ;B02( i , j +1) = B( 2 ) ;B03( i , j +1) = B( 3 ) ;

endend

% This i s the f i l e f o r the three equat ions to be so lved in the Local% Lagrange model .f unc t i on y = legopt (B, j s c1 , j s c2 , j s c3 , A1 , A2 , A3 ,AMax)

y = [B(1) + ( ( j s c 1 − quadgk (@(A) A.ˆ2 . /( exp (A−B(1))−1) ,A1 ,AMax)) ./ ( −0 .25∗ quadgk (@(A) A.ˆ2.∗ ( csch ( (A−B( 1 ) ) / 2 ) ) . ˆ 2 , A1 ,AMax ) ) ) . . .

+ B(2) + ( ( j s c 2 − quadgk (@(A) A.ˆ2 . /( exp (A−B(2))−1) ,A2 , A1)) ./ ( −0 .25∗ quadgk (@(A) A.ˆ2

.∗ ( csch ( (A−B( 2 ) ) / 2 ) ) . ˆ 2 , A2 , A1 ) ) ) . . .+ B(3) + ( ( j s c 3 − quadgk (@(A) A.ˆ2 . /( exp (A−B(3))−1) ,A3 , A2)) ./ ( −0 .25∗ quadgk (@(A) A.ˆ2

.∗ ( csch ( (A−B( 3 ) ) / 2 ) ) . ˆ 2 , A3 , A2 ) ) ) ;j s c 1 − quadgk (@(A) A.ˆ2 . / ( exp (A−B(1))−1) ,A1 ,AMax)− j s c 2 + quadgk (@(A) A.ˆ2 . / ( exp (A−B(2))−1) ,A2 , A1 ) ;

j s c 2 − quadgk (@(A) A.ˆ2 . / ( exp (A−B(2))−1) ,A2 , A1)− j s c 3 + quadgk (@(A) A.ˆ2 . / ( exp (A−B(3))−1) ,A3 , A2 ) ] ;

%%%%%%%%%%%%%%%%%%Global Lagrange Model%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%constant sq=1.6021765e−19; h=6.626069e−34; c =299792458; k=1.38065e−23; f =2.16e−5;

20

Ts=6000; Ta=500; X=1; spectrum =0;

%Fix r e l e v e n t energy gaps and t h e i r d imens i on l e s s parametersEg3 = 0 . 6 6 ;A3 = (Eg3∗q )/ ( k∗Ta ) ;AMax = (22∗q )/ ( k∗Ta ) ;

%Calcu la te Inc iden t EnergySunEnergy = @(A) A.ˆ3 . / ( exp (A∗(Ta/Ts ))−1) ;ps = f ∗ X ∗ quadgk ( SunEnergy , 0 ,AMax) ;

%Choose r ea sonab l e s t a r t i n g energy gaps f o r top two j u n c t i o n s and%reasonab l e p o t e n t i a l s f o r a l l j u n c t i o n sEg1 = 1 . 8 9 ; Eg2 = 1 . 2 ;[Vm1] = henrydbcon (Eg1 , 2 2 ,Ta , spectrum ,X) ;[Vm2] = henrydbcon (Eg2 , Eg1 , Ta , spectrum ,X) ;[Vm3] = henrydbcon (Eg3 , Eg2 , Ta , spectrum ,X) ;

%Make some c o n d i t i o n a l statements f o r dea l i ng with NaN r e s u l t s from%henrydbcon code

TF1=isnan (Vm1) ; TF2=isnan (Vm2) ; TF3=isnan (Vm3) ;i f TF1==1

[Vm1] = dbsjVm(Eg1 , 2 2 ,Ta , spectrum ,X) ;end

i f TF2==1[Vm2] = dbsjVm(Eg2 , Eg1 , Ta , spectrum ,X) ;

endi f TF3==1

[Vm3] = dbsjVm(Eg3 , Eg2 , Ta , spectrum ,X) ;end

%Convert s t a r t i n g po in t s to d imens i on l e s s parametersB01=(q∗Vm1)/( k∗Ta ) ; B02=(q∗Vm2)/( k∗Ta ) ; B03=(q∗Vm3)/( k∗Ta ) ;A1=(Eg1∗q )/ ( k∗Ta ) ; A2=(Eg2∗q )/ ( k∗Ta ) ;

%Solve Legrange Optimizat ion equat ions as coded in l egopt .mB0 = [ B01 , B02 , B03 , A1 , A2 ] ;opt i ons = opt imset ( ’ Display ’ , ’ i t e r ’ , ’ MaxFunEval ’ , 5 0 0 0 , ’ MaxIter ’ , . . .5000 , ’ TolFun ’ , 1 e−9 , ’TolX ’ , 1 e−9);F=@(B) l a g g l o b a l (B, f ,X, Ta , Ts , A3 ,AMax) ;[B, resnom ] =f s o l v e (F , B0 , opt ions ) ;

% This next part i sn ’ t e n t i r e l y nece s sa ry . I put t h i s in only to look at% c e r t a i n va lue s to make sure everyth ing made sense .j 1 = ( f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,B( 4 ) ,AMax) )+ ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,B( 4 ) ,AMax ) ) . . .− quadgk (@(A) A.ˆ2 . / ( exp (A−B(1))−1) ,B( 4 ) ,AMax) ;

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j 2 = ( f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,B( 5 ) ,B( 4 ) ) )+ ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,B( 5 ) ,B ( 4 ) ) ) . . .− quadgk (@(A) A.ˆ2 . / ( exp (A−B(2))−1) ,B( 5 ) ,B( 4 ) ) ;

j 3 = ( f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,A3 ,B( 5 ) ) )+ ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,A3 ,B ( 5 ) ) ) . . .− quadgk (@(A) A.ˆ2 . / ( exp (A−B(3))−1) ,A3 ,B( 5 ) ) ;

j s c 1 = f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,B( 4 ) ,AMax) ;j s c 2 = f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,B( 5 ) ,B( 4 ) ) ;j s c 3 = f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,A3 ,B( 5 ) ) ;jamb1 = ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,B( 4 ) ,AMax) ) ;jamb2 = ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,B( 5 ) ,B( 4 ) ) ) ;jamb3 = ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,A3 ,B( 5 ) ) ) ;j rad1 = −quadgk (@(A) A.ˆ2 . / ( exp (A−B(1))−1) ,B( 4 ) ,AMax) ;j rad2 = −quadgk (@(A) A.ˆ2 . / ( exp (A−B(2))−1) ,B( 5 ) ,B( 4 ) ) ;j rad3 = −quadgk (@(A) A.ˆ2 . / ( exp (A−B(3))−1) ,A3 ,B( 5 ) ) ;

%Determine output power and e f f i c i e n c yp = j1 ∗ (B(1 ) + B(2) + B( 3 ) ) ;E f f = (p/ps )∗100 ;Vm = ( ( k∗Ta)/ q )∗B;

% This i s the s epara te f i l e that conta in s the f i v e equat ions f o r the% g l o b a l model to s o l v e . There i s only f i v e because I have f i x e d the bottom% c e l l to be Ge . Otherwise the re would be 6 .

func t i on y = l a g g l o b a l (B, f ,X, Ta , Ts , A3 ,AMax)

j1 = ( ( f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,B( 4 ) ,AMax) )+ ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,B( 4 ) ,AMax ) ) . . .− quadgk (@(A) A.ˆ2 . / ( exp (A−B(1))−1) ,B( 4 ) ,AMax) ) ;

j1pr ime = (−0.25∗quadgk (@(A) A.ˆ2 .∗ ( csch ( (A−B( 1 ) ) / 2 ) ) . ˆ 2 ,B( 4 ) ,AMax) ) ;j 2 = ( ( f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,B( 5 ) ,B( 4 ) ) )+ ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,B( 5 ) ,B ( 4 ) ) ) . . .− quadgk (@(A) A.ˆ2 . / ( exp (A−B(2))−1) ,B( 5 ) ,B( 4 ) ) ) ;

j2pr ime = (−0.25∗quadgk (@(A) A.ˆ2 .∗ ( csch ( (A−B( 2 ) ) / 2 ) ) . ˆ 2 ,B( 5 ) ,B( 4 ) ) ) ;j 3 = ( ( f ∗X∗quadgk (@(A) A.ˆ2 . / ( exp (A∗(Ta/Ts))−1) ,A3 ,B( 5 ) ) )+ ((1−( f ∗X))∗ quadgk (@(A) A.ˆ2 . / ( exp (A)−1) ,A3 ,B ( 5 ) ) ) . . .− quadgk (@(A) A.ˆ2 . / ( exp (A−B(3))−1) ,A3 ,B( 5 ) ) ) ;

j3pr ime = (−0.25∗quadgk (@(A) A.ˆ2 .∗ ( csch ( (A−B( 3 ) ) / 2 ) ) . ˆ 2 , A3 ,B( 5 ) ) ) ;dj1dA1 = −((( f ∗X∗B(4 )ˆ2 )/ ( exp (B( 4 )∗ (Ta/Ts))−1))+((1−( f ∗X))∗B(4)ˆ2/( exp (B(4))−1))−(B(4)ˆ2/( exp (B(4)−B(1) ) −1 ) ) ) ;dj2dA1 = ( ( ( f ∗X∗B(4 )ˆ2 )/ ( exp (B( 4 )∗ (Ta/Ts))−1))+((1−( f ∗X))∗B(4)ˆ2/( exp (B(4))−1))−(B(4)ˆ2/( exp (B(4)−B(2) ) −1 ) ) ) ;dj2dA2 = −((( f ∗X∗B(5 )ˆ2 )/ ( exp (B( 5 )∗ (Ta/Ts))−1))+((1−( f ∗X))∗B(5)ˆ2/( exp (B(5))−1))−(B(5)ˆ2/( exp (B(5)−B(2) ) −1 ) ) ) ;dj3dA2 = ( ( ( f ∗X∗B(5 )ˆ2 )/ ( exp (B( 5 )∗ (Ta/Ts))−1))+((1−( f ∗X))∗B(5)ˆ2/( exp (B(5))−1))−(B(5)ˆ2/( exp (B(5)−B(3) ) −1 ) ) ) ;

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y = [B(1)+B(2)+B(3)+( j1 / j1pr ime )+( j2 / j2pr ime )+( j3 / j3pr ime ) ;j1−j 2 ;j2−j 3 ;dj1dA1 ∗( j 1 / j1pr ime ) + dj2dA1 ∗ ( ( j 2 / j2pr ime ) ) ;dj2dA2 ∗ ( ( j 2 / j2pr ime ) ) + dj3dA2 ∗( j 3 / j3pr ime ) ] ;

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