1148:, because radiative recombination cannot be avoided other than by avoiding light absorption (principle of detailed balance). However, since absorption is a key requirement for a solar cell and necessary to achieve a high concentration of electrons and holes as well, radiative recombination is a necessity (see van Roosbroeck-Shockley equation ). If non-radiative recombination is substantial and non negligible, the open-circuit voltage will be reduced depending on the ratio between the radiative and non-radiative recombination currents (where the recombination currents are the integral of the recombination rates over volume). This leads to a second reciprocity relation between the photovoltaic and the luminescent operation mode of a solar cell because the ratio of radiative to total (radiative and non-radiative) recombination currents is the external luminescence quantum efficiency
81:. A solar cell and a light emitting diode are typically made from different materials and optimized for different purposes; however, conceptually every solar cell could be operated as a light emitting diode and vice versa. Given that the operation principles have a high symmetry it is fair to assume that the key figures of merit that are used to characterize photovoltaic and luminescent operation of diodes are related to each other. These relations become particularly simple in a situation, where recombination rates scale linearly with minority carrier density and are explained below.
819:, the collection efficiency is one. Further away from the edge of the space charge region, the collection efficiency will be smaller than one depending on the distance and the amount of recombination happening in the neutral zone. The same holds for the electron concentration in the dark under applied bias. Here, the electron concentration will also decrease from the edge of the space charge region towards the back contact. This decrease as well as the collection efficiency will be approximately exponential (with the diffusion length controlling the decay).
90:
104:
quantum efficiency of photocurrent in the photovoltaic situation on the left. The relation between the two situations is based on the principle of detailed balance that relates absorption and radiative recombination via the van
Roosbroeck-Shockley equation and charge collection and injection via the Donolato theorem.
546:
Equation (1) is valid for the practically relevant situation, where the neutral base region of a pn-junction makes up most of the volume of the diode. Typically, the thickness of a crystalline Si solar cell is ~ 200 μm while the thickness of the emitter and space charge region is only on the
826:
and connects the processes of charge carrier injection (relevant in the luminescent mode of operation) and charge carrier extraction (relevant in the photovoltaic mode of operation). In addition, the detailed balance between absorption of photons and radiative recombination can be mathematically
542:
is the temperature of the diode. This simple relation is useful for the analysis of solar cells using luminescence-based characterization methods. Luminescence used for characterization of solar cells is useful because of the ability to image the luminescence of solar cells and modules in short
103:
between the n and p-type regions of the diode. On the right, a forward voltage is applied to the same diode. Electron injection will lead to recombination and consequently light emission. The emission spectrum of the luminescence emitted in the situation on the right is directly related to the
1049:
solar cells. The reciprocity relation is also invalid if the emission of the solar cell is not from delocalized conduction and valence band states as would be the case for most mono and polycrystalline semiconductors but from localized states (defect states). This limitation is relevant for
1101:
of the solar cell and does not change much between different devices of the same type. The rate of recombination however might vary over orders of magnitude depending on the quality of the material and the interfaces. Thus, the open-circuit voltage depends quite drastically on the rates of
547:
order of hundreds of nanometers, i.e. three orders of magnitude thinner. In the base of a pn-junction, recombination is typically linear with minority carrier concentration over a large range of injection conditions and charge carrier transport is by
60:
that are operated in a different voltage and illumination regime and that serve different purposes. A solar cell is operated under illumination (usually by solar radiation) and is typically kept at the maximum power point where the product of
1088:
of a solar cell is the voltage created by a certain amount of illumination if the contacts of the solar cell are not connected, i.e. in open circuit. The voltage that can build up in such as situation is directly connected to the
1294:
457:
93:
Illustration of the basic underlying principles of the reciprocity relation between photovoltaic quantum efficiency and external luminescence quantum efficiency of a light emitting diode. On the left, the band diagram of a
98:
solar cell is depicted with a thin n-type region on the left and a thicker p-type region on the right. Light absorption in the p-type base leads to free electrons that have to be collected by diffusing to the edge of the
916:
734:
1400:
1029:
The reciprocity relation (eq. (1)) is only valid if absorption and emission is dominated by the neutral region of the pn-junction shown in the adjacent figure. This is a good approximation for
1579:
Kirchartz, T.; Nelson, J.; Rau, U. (2016). "Reciprocity between charge injection and extraction and its influence on the interpretation of electroluminescence spectra in organic solar cells".
1614:
Müller, T. C. M.; Pieters, B. E.; Kirchartz, T.; Carius, R.; Rau, U. (2014). "Effect of localized states on the reciprocity between quantum efficiency and electroluminescence in Cu(In,Ga)Se
641:
305:
1334:
543:
periods of times, while spatially resolved measurements of photovoltaic properties (such as photocurrent or photovoltage) would be very time-consuming and technically difficult.
65:
and voltage are maximized. A light emitting diode is operated at an applied forward bias (without external illumination). While a solar cell converts the energy contained in the
966:
1086:
1097:. The rate of photogeneration is usually determined by the typically used illumination with white light with a power density of 100 mW/cm (called one sun) and by the
576:
341:
is the black body spectrum emitted by a surface (the diode) into the hemisphere above the diode in units of photons per area, time and electron interval. In this case the
1142:
339:
179:
1185:
145:
939:
817:
1024:
993:
784:
1762:
Rau, U.; Blank, B.; Müller, T. C. M.; Kirchartz, T. (2017). "Efficiency potential of photovoltaic materials and devices unveiled by detailed-balance analysis".
757:
540:
520:
500:
480:
199:
1299:
Thus, any reduction in the external luminescence quantum efficiency by one order of magnitude would lead to a reduction in open-circuit voltage (relative to
1684:
Vandewal, K.; Tvingstedt, K.; Gadisa, A.; Inganas, O.; Manca, J. V. (2009). "On the origin of the open-circuit voltage of polymer-fullerene solar cells".
1193:
351:
1034:
833:
1402:. Equation (2) is frequently used in the literature on solar cells. For instance for an improved understanding of the open-circuit voltage in
649:
1094:
35:
1102:
recombination at a given concentration of charge carriers. The highest possible open-circuit voltage, the radiative open-circuit voltage
34:. The relations are useful for interpretation of luminescence based measurements of solar cells and modules and for the analysis of
786:
defines the position of the edge of the space charge region (where the neutral zone and the space charge region connect). Thus, if
1339:
201:. Under certain conditions specified below, these two properties measured on the same diode are connected via the equation
147:
is a spectral quantity that is generally measured as a function of photon energy (or wavelength). The same is true for the
581:
207:
85:
Reciprocity between the photovoltaic quantum efficiency and the electroluminescence spectrum of a pn-junction diode
1302:
78:
66:
1803:
1505:
van
Roosbroeck, W.; Shockley, W. (1954). "Photon-radiative recombination of electrons and holes in germanium".
1430:"Reciprocity relation between photovoltaic quantum efficiency and electroluminescent emission of solar cells"
1054:
Reciprocity between the open-circuit voltage of a solar cell and the external luminescence quantum efficiency
1145:
1144:, is obtained if all recombination is radiative and non-radiative recombination is negligible. This is the
1771:
1693:
1658:
1588:
1514:
1476:
1441:
1059:
342:
77:(voltage x current) a light-emitting diode does the inverse, namely converting electrical power into
50:
944:
1064:
1038:
1030:
1026:
is the intrinsic charge carrier concentration. A derivation of equation (1) can be found in ref.
148:
100:
554:
1744:
1561:
1403:
1042:
109:
1105:
314:
154:
1151:
551:. In this situation, the Donolato theorem. is valid that states that the collection efficiency
1709:
1429:
1046:
114:
924:
789:
1779:
1736:
1701:
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1627:
1596:
1553:
1522:
1484:
1449:
996:
823:
62:
1002:
971:
762:
70:
1095:
photogeneration (determined by the amount of illumination) and the rates of recombination
1775:
1697:
1662:
1592:
1542:"On the Use of Rau's Reciprocity to Deduce External Radiative Efficiency in Solar Cells"
1518:
1480:
1445:
1090:
742:
525:
505:
485:
465:
184:
74:
1041:
is of comparable size to the total absorber volume. This is the case for instance for
57:
1797:
54:
1783:
1748:
1727:
Green, M. A. (2012). "Radiative efficiency of state-of-the-art photovoltaic cells".
1600:
1565:
1289:{\displaystyle qV_{{\text{oc}},{\text{rad}}}-qV_{oc}=-kT\ln {Q_{e,{\text{lum}}}}(2)}
452:{\displaystyle \phi _{bb}={\frac {2\pi }{h^{3}c^{2}}}{\frac {E^{2}}{\exp {E/kT}-1}}}
95:
1631:
1557:
89:
1646:
1453:
1406:
and for comparing voltage losses between different photovoltaic technologies.
1037:. However the equations has limitations when applied to solar cells where the
46:
1526:
1187:
of a (light emitting) diode. Mathematically, this relation is expressed as,
548:
1713:
1541:
911:{\displaystyle k_{\text{rad}}n_{i}^{2}=\int \alpha 4n_{r}^{2}\phi _{bb}dE}
1098:
729:{\displaystyle f_{\text{c}}(x)={\frac {\delta n(x)}{\delta n(x=x_{j})}}}
31:
1670:
1740:
1705:
27:
1488:
1467:
Donolato, C. (1985). "A reciprocity theorem for charge collection".
88:
23:
1093:
in the device. These densities in turn depend on the rates of
578:
is related to the normalized minority carrier concentration
1395:{\displaystyle kT/q\times \ln(10)\approx \mathrm {60~mV} }
827:
expressed using the van
Roosbroeck–Shockley equation as
181:
of a light emitting diode under applied forward voltage
1342:
1305:
1196:
1154:
1108:
1067:
1005:
974:
947:
927:
836:
792:
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745:
652:
584:
557:
528:
508:
488:
468:
354:
317:
210:
187:
157:
117:
1050:
microcrystalline and amorphous silicon solar cells.
1394:
1328:
1288:
1179:
1136:
1080:
1018:
987:
960:
933:
910:
822:The Donolato theorem is based on the principle of
811:
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728:
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570:
534:
514:
494:
474:
451:
333:
299:
193:
173:
139:
1423:
1421:
1419:
1033:solar cells and the method can also be used for
1647:"Some thermodynamics of photochemical systems"
636:{\displaystyle \delta n(x)/\delta n(x=x_{j})}
300:{\displaystyle \phi _{EL}=Q_{e,PV}\phi _{bb}}
8:
1329:{\displaystyle V_{{\text{oc}},{\text{rad}}}}
968:is the radiative recombination coefficient,
1035:copper indium gallium selenide solar cells
1378:
1349:
1341:
1319:
1311:
1310:
1304:
1269:
1262:
1257:
1230:
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116:
30:emission of the same diode under applied
1415:
1620:Solar Energy Materials and Solar Cells
1500:
1498:
522:is the speed of light in vacuum, and
16:Relation between properties of diodes
7:
20:Optoelectronic reciprocity relations
1540:Wang, X.; Lundstrom, M. S. (2013).
1388:
1385:
14:
1784:10.1103/physrevapplied.7.044016
1618:and Si thin-film solar cells".
1601:10.1103/physrevapplied.5.054003
941:is the absorption coefficient,
1372:
1366:
1283:
1277:
1091:density of electrons and holes
961:{\displaystyle k_{\text{rad}}}
720:
701:
690:
684:
669:
663:
630:
611:
597:
591:
294:
259:
1:
1546:IEEE Journal of Photovoltaics
1081:{\displaystyle V_{\text{oc}}}
1632:10.1016/j.solmat.2014.04.018
1558:10.1109/jphotov.2013.2278658
759:is a spatial coordinate and
571:{\displaystyle f_{\text{c}}}
482:is the Boltzmann constant,
1820:
1454:10.1103/physrevb.76.085303
1137:{\displaystyle V_{oc,rad}}
334:{\displaystyle \phi _{bb}}
174:{\displaystyle \phi _{EL}}
26:under illumination to the
1180:{\displaystyle Q_{e,lum}}
79:electromagnetic radiation
502:is the Planck constant,
140:{\displaystyle Q_{e,PV}}
1764:Physical Review Applied
1581:Physical Review Applied
1527:10.1103/physrev.94.1558
1469:Applied Physics Letters
934:{\displaystyle \alpha }
812:{\displaystyle x=x_{j}}
38:losses in solar cells.
22:relate properties of a
1396:
1330:
1290:
1181:
1138:
1082:
1020:
989:
962:
935:
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141:
105:
1397:
1331:
1291:
1182:
1139:
1083:
1021:
1019:{\displaystyle n_{i}}
990:
988:{\displaystyle n_{r}}
963:
936:
913:
814:
781:
779:{\displaystyle x_{j}}
754:
731:
638:
573:
537:
517:
497:
477:
454:
336:
302:
196:
176:
142:
92:
67:electromagnetic waves
51:light-emitting diodes
1645:Ross, R. T. (1967).
1340:
1303:
1194:
1152:
1106:
1065:
1060:open-circuit voltage
1003:
972:
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834:
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315:
208:
185:
155:
115:
1776:2017PhRvP...7d4016R
1698:2009NatMa...8..904V
1663:1967JChPh..46.4590R
1593:2016PhRvP...5e4003K
1519:1954PhRv...94.1558V
1481:1985ApPhL..46..270D
1446:2007PhRvB..76h5303R
1404:organic solar cells
1043:organic solar cells
1039:space charge region
1031:crystalline silicon
888:
861:
343:black body spectrum
149:electroluminescence
101:space charge region
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110:quantum efficiency
106:
1671:10.1063/1.1840606
1657:(12): 4590–4593.
1434:Physical Review B
1384:
1322:
1314:
1272:
1216:
1208:
1075:
955:
844:
752:{\displaystyle x}
724:
660:
565:
535:{\displaystyle T}
515:{\displaystyle c}
495:{\displaystyle h}
475:{\displaystyle k}
447:
403:
286:
194:{\displaystyle V}
108:The photovoltaic
1811:
1788:
1787:
1759:
1753:
1752:
1741:10.1002/pip.1147
1724:
1718:
1717:
1706:10.1038/nmat2548
1686:Nature Materials
1681:
1675:
1674:
1642:
1636:
1635:
1611:
1605:
1604:
1576:
1570:
1569:
1552:(4): 1348–1353.
1537:
1531:
1530:
1513:(6): 1558–1560.
1502:
1493:
1492:
1464:
1458:
1457:
1428:Rau, U. (2007).
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997:refractive index
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824:detailed balance
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69:of the incoming
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1804:Optoelectronics
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1729:Prog. Photovolt
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1507:Physical Review
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1146:ideal situation
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71:solar radiation
44:
17:
12:
11:
5:
1817:
1815:
1807:
1806:
1796:
1795:
1790:
1789:
1754:
1735:(4): 472–476.
1719:
1676:
1637:
1615:
1606:
1571:
1532:
1494:
1475:(3): 270–272.
1459:
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75:electric power
55:semiconducting
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3:
2:
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1770:(4): 044016.
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1692:(11): 904–9.
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1652:
1651:J. Chem. Phys
1648:
1641:
1638:
1633:
1629:
1625:
1621:
1610:
1607:
1602:
1598:
1594:
1590:
1587:(5): 054003.
1586:
1582:
1575:
1572:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1536:
1533:
1528:
1524:
1520:
1516:
1512:
1508:
1501:
1499:
1495:
1490:
1486:
1482:
1478:
1474:
1470:
1463:
1460:
1455:
1451:
1447:
1443:
1440:(8): 085303.
1439:
1435:
1431:
1424:
1422:
1420:
1416:
1409:
1407:
1405:
1379:
1375:
1369:
1363:
1360:
1357:
1354:
1350:
1346:
1343:
1316:
1307:
1280:
1266:
1263:
1259:
1254:
1251:
1248:
1245:
1242:
1239:
1234:
1231:
1227:
1223:
1220:
1210:
1201:
1197:
1190:
1189:
1188:
1172:
1169:
1166:
1163:
1160:
1156:
1147:
1129:
1126:
1123:
1120:
1117:
1114:
1110:
1100:
1096:
1092:
1069:
1061:
1053:
1051:
1048:
1044:
1040:
1036:
1032:
1027:
1011:
1007:
998:
980:
976:
949:
928:
905:
902:
897:
894:
890:
884:
879:
875:
871:
868:
865:
862:
857:
852:
848:
838:
830:
829:
828:
825:
820:
804:
800:
796:
793:
771:
767:
746:
715:
711:
707:
704:
698:
695:
687:
681:
678:
672:
666:
654:
646:
645:
644:
625:
621:
617:
614:
608:
605:
601:
594:
588:
585:
559:
550:
544:
529:
509:
489:
469:
443:
440:
436:
433:
429:
425:
421:
418:
412:
408:
397:
393:
387:
383:
377:
374:
368:
363:
360:
356:
348:
347:
346:
344:
326:
323:
319:
291:
288:
282:
279:
274:
271:
265:
262:
254:
251:
247:
241:
238:
235:
232:
228:
224:
219:
216:
212:
204:
203:
202:
188:
166:
163:
159:
150:
132:
129:
126:
123:
119:
111:
102:
97:
91:
84:
82:
80:
76:
72:
68:
64:
59:
56:
52:
48:
41:
39:
37:
36:recombination
33:
29:
25:
21:
1767:
1763:
1757:
1732:
1728:
1722:
1689:
1685:
1679:
1654:
1650:
1640:
1623:
1619:
1609:
1584:
1580:
1574:
1549:
1545:
1535:
1510:
1506:
1472:
1468:
1462:
1437:
1433:
1298:
1057:
1047:amorphous Si
1028:
920:
821:
738:
545:
461:
345:is given by
310:
107:
96:p-n junction
45:
19:
18:
47:Solar cells
1626:: 95–103.
1410:References
1376:≈
1364:
1358:×
1255:
1243:−
1221:−
929:α
891:ϕ
869:α
866:∫
696:δ
679:δ
606:δ
586:δ
549:diffusion
441:−
422:
378:π
357:ϕ
320:ϕ
289:−
266:
248:ϕ
213:ϕ
160:ϕ
151:spectrum
53:are both
1798:Category
1749:94696623
1714:19820700
1566:24481366
1099:band gap
1772:Bibcode
1694:Bibcode
1659:Bibcode
1589:Bibcode
1515:Bibcode
1477:Bibcode
1442:Bibcode
995:is the
63:current
32:voltage
1747:
1712:
1564:
1383:
921:Here,
739:where
462:where
311:where
58:diodes
42:Basics
28:photon
1745:S2CID
1562:S2CID
1336:) by
73:into
24:diode
1710:PMID
1058:The
643:via
49:and
1780:doi
1737:doi
1702:doi
1667:doi
1628:doi
1624:129
1597:doi
1554:doi
1523:doi
1485:doi
1450:doi
1321:rad
1271:lum
1215:rad
1045:or
999:,
954:rad
843:rad
419:exp
307:(1)
263:exp
1800::
1778:.
1766:.
1743:.
1733:20
1731:.
1708:.
1700:.
1688:.
1665:.
1655:46
1653:.
1649:.
1622:.
1595:.
1583:.
1560:.
1548:.
1544:.
1521:.
1511:94
1509:.
1497:^
1483:.
1473:46
1471:.
1448:.
1438:76
1436:.
1432:.
1418:^
1380:60
1370:10
1361:ln
1313:oc
1252:ln
1207:oc
1074:oc
1786:.
1782::
1774::
1768:7
1751:.
1739::
1716:.
1704::
1696::
1690:8
1673:.
1669::
1661::
1634:.
1630::
1616:2
1603:.
1599::
1591::
1585:5
1568:.
1556::
1550:3
1529:.
1525::
1517::
1491:.
1487::
1479::
1456:.
1452::
1444::
1389:V
1386:m
1373:)
1367:(
1355:q
1351:/
1347:T
1344:k
1317:,
1308:V
1284:)
1281:2
1278:(
1267:,
1264:e
1260:Q
1249:T
1246:k
1240:=
1235:c
1232:o
1228:V
1224:q
1211:,
1202:V
1198:q
1173:m
1170:u
1167:l
1164:,
1161:e
1157:Q
1130:d
1127:a
1124:r
1121:,
1118:c
1115:o
1111:V
1070:V
1012:i
1008:n
981:r
977:n
950:k
906:E
903:d
898:b
895:b
885:2
880:r
876:n
872:4
863:=
858:2
853:i
849:n
839:k
805:j
801:x
797:=
794:x
772:j
768:x
747:x
721:)
716:j
712:x
708:=
705:x
702:(
699:n
691:)
688:x
685:(
682:n
673:=
670:)
667:x
664:(
659:c
655:f
631:)
626:j
622:x
618:=
615:x
612:(
609:n
602:/
598:)
595:x
592:(
589:n
564:c
560:f
530:T
510:c
490:h
470:k
444:1
437:T
434:k
430:/
426:E
413:2
409:E
398:2
394:c
388:3
384:h
375:2
369:=
364:b
361:b
327:b
324:b
295:]
292:1
283:T
280:k
275:V
272:q
260:[
255:b
252:b
242:V
239:P
236:,
233:e
229:Q
225:=
220:L
217:E
189:V
167:L
164:E
133:V
130:P
127:,
124:e
120:Q
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