Detailed balance limit of the efficiency of multilevel intermediate bandsolar cells

Tomohiro Nozawa1,2 and Yasuhiko Arakawa1,3,a�

1Institute for Nano Quantum Information Electronics, The University of Tokyo, Komaba, Meguro-ku,Tokyo 153-8505, Japan2Advanced Technology Research Laboratories, Sharp Corporation, 2613-1 Ichinomoto-cho, Tenri,Nara 632-8567, Japan3Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan

�Received 28 December 2010; accepted 30 March 2011; published online 27 April 2011�

Intermediate-band solar cells �IBSCs� promise ultrahigh solar-electricity energy conversion. Wehave calculated the detailed balance limit of the efficiency for IBSCs with multiple intermediatebands by optimizing IB’s energy levels. The results indicate that thermodynamic limit of IBSCs with4 IBs is 74.6% which far exceeds 63% calculated in a previous study for the single IB case. Byfurther increasing the total number IBs, the thermodynamic limit of IBSCs can ultimately approachnearly 80%. © 2011 American Institute of Physics. �doi:10.1063/1.3583587�

Intermediate-band solar cells �IBSCs� are a promisingtechnology for realizing ultrahigh efficiency of solar energyconversion. In 1997, Luque and Martí1 predicted a 63% ef-ficiency as the thermodynamic upper limit in IBSCs relativeto 41% for conventional single-junction cells under fullconcentration, i.e., 46 000 suns. This intermediate-band ap-proach has been intensively studied these days by experi-mentally adopting quantum well2,3 and quantum dot4,5 struc-tures in photovoltaic active layers.

Luque et al.6 recently applied a detailed balance limitcalculation model to investigate IBSCs for their experimentalsample solar cells. They calculated the efficiency of an IBSCwith four intermediate bands �IBs� for a matrix band gap andIBs’ energy levels measured. Their calculation showed anefficiency of 40.0%, which is smaller than the thermody-namic upper limit efficiency of 63% for an IBSC with asingle IB. The reason for this poor efficiency is that Eg andIBs’ energy levels are unoptimized. Green reported on thegeneral theory of impurity photovoltaic solar cells with mul-tiple bands �N-band cells� and mentioned that the perfor-mance limit of an N-band cell can approach that of anN-band tandem as N approaches infinity. However, the effi-ciency of an N-band cell was not calculated.7 In this letter,we discuss the detailed balance limit of the efficiencies forIBSCs with multiple IBs, calculating the efficiencies by op-timizing Eg and IB energy levels for each case of variousnumbers of IBs. We show that the maximum theoretical ef-ficiency of the IBSCs is well above 63% obtained in Ref. 1,approaching 80% by increasing the number of intermediatelevels.

The detailed balance limit represents the thermodynamicenergy conversion efficiency limit of solar cells accountingfor black-body radiation. �See Refs. 8 and 9 for details of thetheory and formalism.� We followed the calculation condi-tions in Ref. 1. In the following the calculation method forIBSCs with 4 IBs �i.e., 6-level IBSCs� is explained as anexample. A schematic diagram of carrier transitions betweenone energy level and another is depicted in Fig. 1�a�, whereCB and VB are the conduction and valence band, respec-

tively. Inter IB transitions have spectra with infinitesimallinewidths and absorbed photon fluxes between IBs are muchsmaller compared to the other transitions shown in Fig. 1�a�.For this reason, those transitions were ignored. Solid anddotted arrows represent carrier generation and recombina-tion, respectively. Figure 1�b� shows energy gaps Ei betweenVB and IBi �i=1,2 ,3 ,4� and that of matrix semiconductorEg. It should be noted that more than one set of Ei’s result ina same energy conversion efficiency for a given Eg due tosymmetrical equivalency between the first excitation fromVB to Ei and the second from Ei to CB. We assume thatinterband transitions have no overlap for wavelengths of thephoton fluxes one another as Luque et al. supposed.1,6

Namely, we divide the whole incident solar spectrum intonine parts �double the number of IBs+1� in wavelengths andeach wavelengths component contributes only to a singleinterband transition. We assume that the photon flux withenergies between Eini and Efin �Efin�Eini� from a matterobeys Planck radiation and is given by

N�T,�,Eini,Efin� =2�

h3c2�Eini

Efin E2

exp�E − �

kT� − 1

dE , �1�

where T is the temperature of the matter, � is the photonchemical potential equal to the separation of quasi-Fermi

a�Electronic mail: arakawa@iis.u-tokyo.ac.jp.FIG. 1. Schematic diagram of �a� carrier transitions and energy levels and�b� energy gaps.

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levels, h is the Planck’s constant, c is the speed of light, k isthe Boltzman constant. Using Eq. �1�, the photon fluxes ofGCV, RCV, GVI4, and RVI4 are written as

GCV = C0HN�Ts,0,Eg,�� , �2�

RCV = N�T0,qV,Eg,�� , �3�

GVI4 = C0HN�Ts,0,E4,Eg� , �4�

RVI4 = N�T0,�5,E4,Eg� , �5�

where C0 is the concentration ratio �C0 suns illumination�, Tsis the sun temperature, T0 is the solar cell temperature, H issin2 �s, �s is the sun’s semiangle of vision determined by thesun’s radius and its distance to the Earth, 0.267°.

Other photon fluxes are described similarly. No carrier isextracted from any IB, which is mathematically expressed as

0 = GVI1 − GCI1 − RVI1 + RCI1, �6�

0 = GVI2 − GCI2 − RVI2 + RCI2, �7�

0 = GVI3 − GCI3 − RVI3 + RCI3, �8�

0 = GVI4 − GCI4 − RVI4 + RCI4. �9�

The solar cell output voltage is the sum of the five chemicalpotentials �� j for j=1,2 ,3 ,4 ,5�. Therefore, the chemical po-tential � j can be obtained from Eqs. �6�–�9� for a given qV.With these parameters the current output of the solar cell is

calculated. The current-voltage characteristic of the solar cellis described as

J

q= GCV + GCI1 + GCI2 + GCI3 + GCI4 − RCV − RCI1 − RCI2

− RCI3 − RCI4. �10�

From this equation the maximum power of JV and the effi-ciency of the solar cell are obtained. By the calculationscheme above, detailed balance limits of the efficiencies forIBSCs with 0, 1, 2, 3, and 4 IBs were simulated for Ts of6000 K and T0 of 300 K. Each IBSC with 0, 1, 2, 3, and 4IBs has the number of total �CB+VB+IBs� energy levels/bands �Ntotal� of 2, 3, 4, 5, and 6, and is named as No-IB solarcell, 3-, 4-, 5-, and 6-level IBSC, respectively. Figure 2shows the calculation results of limit efficiencies for varioustypes of solar cells as a function of the bandgap of matrixsemiconductor, Eg, under �a� full concentration and �b� noconcentration, with IBs optimized to maximize the limit ef-ficiency for each Eg. It should be noted that our calculationresults for 3-level IBSC under full concentration shown inFig. 2�a� are quite consistent with those obtained in Ref. 1.From this figure it can be seen that the limit efficienciessignificantly increase with increasing Ntotal except for smallEg’s, especially below approximately 1.2 eV, where the effi-ciencies of IBSCs are almost independent on Ntotal. The in-crease in limit efficiency in this region is insensitive to Eg,which we interpret as resulting from the smaller potential for

FIG. 2. �Color online� Calculated detailed balance limit of the efficienciesfor various types of solar cells as a function of the bandgap energy of matrixsemiconductor, Eg, under �a� full concentration and �b� no concentration. IBenergies, Ei’s, are optimized to maximize the efficiency for each Eg.

FIG. 3. �Color online� Calculated variation in optimal Eg’s to maximize thelimit efficiency for each Ntotal under varied sunlight concentrations.

FIG. 4. �Color online� Calculated detailed balance limit of the efficiencies asa function of Ntotal for each sunlight concentration. Eg and Ei are optimizedto maximize the efficiency for each Ntotal and sunlight concentration.

171108-2 T. Nozawa and Y. Arakawa Appl. Phys. Lett. 98, 171108 �2011�

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current increase for low Eg’s. On the other hand, higher Eg’sgive larger efficiency increment with increased Ntotal andover 10% increase in efficiencies for 4-level IBSC relative to3-level IBSC is observed for a wide range of Eg above 2.0eV for instance.

Figure 3 gives the variation in optimal Eg’s to maximizethe limit efficiency depending on Ntotal under varied sunlightconcentrations. The optimal Eg’s dramatically increase fromNo-IB solar cell to 3-level IBSC, while showing relativelygradual evolution from 3 to 6-level IBSC. The optimal Eg’sare found to be in small ranges for multilevel IBSCs and donot vary much by the difference of the IB number for eachsunlight concentration ratios. The optimal Eg’s decrease withincreasing the concentration ratio due to mitigation of Vmaxoffset from Eg by increased photocurrent.

In Fig. 4, we summarize the limit efficiencies dependingon Ntotal for each sunlight concentration. It can be seen fromthis figure that the limit efficiency under full concentrationapproaches 80% with increasing Ntotal.

For each sunlight concentration in each type of solarcells, the limit efficiencies, optimal Eg’s and Ei’s are summa-rized in Table I. Again, it should be noted that more than oneset of Ei’s result in a same energy conversion efficiency for agiven Eg due to symmetrical equivalency between the firstexcitation from VB to Ei and the second from Ei to CB. Weshow only one set of Ei’s for each Eg in descending order inthis table. From this table, we notice that the limit efficiencyfor 6-level IBSC reaches approximately 75% under full con-centration. Even for 4-level IBSC, the limit efficiency is cal-culated to be as high as 70% under full concentration, 64%under 1000 suns and 53% without concentration. Further-more, the limit efficiency of over 77% is obtained for anIBSC with 15 IBs under full concentration.

In conclusion, we have calculated detailed balance limitof the efficiencies for IBSCs with increased numbers of IBenergy levels. The limit efficiency of a 6-level IBSC �i.e.,IBSC with four IBs� is found to be 74.6% and approaches80% by further addition of IBs. These limit efficiencies farexceed the 63% calculated in Ref. 1 for an IBSC with asingle IB �i.e., 3-level IBSC�. This work is a strong implica-tion of the potential of multilevel IBSCs to realize ultrahighefficiency.

The authors would like to thank S. Iwamoto and M.Kitamura for their useful discussions and K. Tanabe for hisinvaluable comments on our manuscript as well as usefuldiscussions. The authors also would like to acknowledge A.Takahashi, Y. Tomomura, S. Aomori, and M. Izumi for theirencouragement and support to this work. This work was sup-ported in part by the Special Coordination Funds for Promot-ing Science and Technology by the Ministry of Education,Culture, Sports, Science and Technology �MEXT�, Japan.

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TABLE I. Limit efficiencies, optimal Eg’s and Ei’s for each sunlight con-centration in each type of solar cells. Note that more than one set of Ei’sresult in a same energy conversion efficiency for a given Eg due to sym-metrical equivalency between the first excitation from VB to Ei and thesecond from Ei to CB. We show only one set of Ei’s for each Eg in descend-ing order in this table.

No concentration

Efficiency�%�

Eg

�eV�E4

�eV�E3

�eV�E2

�eV�E1

�eV�

No-IB solar cell 31.0 1.30 ¯ ¯ ¯ ¯

3-level IBSC 46.8 2.39 1.48 ¯ ¯ ¯

4-level IBSC 53.0 2.58 1.80 1.44 ¯ ¯

5-level IBSC 55.5 2.63 1.93 1.63 1.41 ¯

6-level IBSC 56.8 2.68 1.99 1.73 1.55 1.41

1000 suns

Efficiency�%�

Eg

�eV�E4

�eV�E3

�eV�E2

�eV�E1

�eV�No-IB solar cell 37.1 1.19 ¯ ¯ ¯ ¯

3-level IBSC 57.3 2.09 1.32 ¯ ¯ ¯

4-level IBSC 63.8 2.30 1.65 1.30 ¯ ¯

5-level IBSC 66.5 2.38 1.81 1.51 1.29 ¯

6-level IBSC 67.9 2.44 1.90 1.63 1.44 1.29

Full concentration

Efficiency�%�

Eg

�eV�E4

�eV�E3

�eV�E2

�eV�E1

�eV�No-IB solar cell 40.7 1.10 ¯ ¯ ¯ ¯

3-level IBSC 63.2 1.95 1.24 ¯ ¯ ¯

4-level IBSC 70.1 2.19 1.59 1.24 ¯ ¯

5-level IBSC 73.1 2.25 1.73 1.44 1.23 ¯

6-level IBSC 74.6 2.29 1.82 1.56 1.38 1.22

171108-3 T. Nozawa and Y. Arakawa Appl. Phys. Lett. 98, 171108 �2011�

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