OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 16927–16933
« Show journal navigation

Detailed balance model for intermediate band solar cells with photon conservation

Chien-chung Lin, Wei-Ling Liu, and Ching-Yu Shih  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 16927-16933 (2011)
http://dx.doi.org/10.1364/OE.19.016927


View Full Text Article

Acrobat PDF (877 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We developed a comprehensive detailed balance model of intermediate band solar cell (IBSC). The key feature of our model is based on the conservation of photons in solar spectrum. Together with parametric analysis of carrier partition, we calculated the power conversion efficiency and found an enhancement of 1.5 times in wide band gap material IBSC (such as GaN). On the other hand, this model can also explain the inferior performance of GaAs-based IBSC through the degradation of open-circuit voltages, which can be attributed to the strong non-radiative recombination and the increased photo-generated carriers. The resulting maximum efficiency is complied with the classical Shockley-Queisser limit, and should be considered for the future IBSC design.

© 2011 OSA

1. Introduction

The intermediate band solar cells (IBSCs) have attracted great attentions due to the possibility of exceeding the Shockley-Queisser (SQ) limit [1

1. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]

]. From the analysis by Luque et al. [2

2. A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]

], the ideal efficiency can be as high as 62%. Since its introduction in 1997, there have been many efforts, including theoretical and experimental works, to explore this idea. In many cases, quantum dot (QD) solar cells were proposed as the possible candidate for IBSC implementation [3

3. T. Sugaya, S. Furue, H. Komaki, T. Amano, M. Mori, K. Komori, S. Niki, O. Numakami, and Y. Okano, “Highly stacked and well-aligned In0.4Ga0.6As quantum dot solar cells with In0.2Ga0.8As cap layer,” Appl. Phys. Lett. 97(18), 183104 (2010). [CrossRef]

5

5. R. Oshima, A. Takata, Y. Shoji, K. Akahane, and Y. Okada, “InAs/GaNAs strain-compensated quantum dots stacked up to 50 layers for use in high-efficiency solar cell,” Physica E 42(10), 2757–2760 (2010). [CrossRef]

]. However, so far, the expected superior performance of IBSC has not been realized in a practical manner. In most cases, the demonstrated power conversion efficiencies were either inferior or at most equivalent to their single junction counterparts. Several explanations or modifications have been addressed: separation of quasi-Fermi level (QFL) [6

6. G. D. Wei, K.-T. Shiu, N. C. Giebink, and S. R. Forrest, “Thermodynamic limits of quantum photo-voltaic cell efficiency,” Appl. Phys. Lett. 91(22), 223507 (2007). [CrossRef]

], absorption between intermediate band [7

7. W. G. Hu, T. Inoue, O. Kojima, and T. Kita, “Effects of absorption coefficients and intermediate-band filling in InAs/GaAs quantum dot solar cells,” Appl. Phys. Lett. 97(19), 193106 (2010). [CrossRef]

], occupation rate of IB [8

8. K. Yoshida, Y. Okada, and N. Sano, “Self-consistent simulation of intermediate band solar cells: Effect of occupation rates on device characteristics,” Appl. Phys. Lett. 97(13), 133503 (2010). [CrossRef]

], and low threshold Auger generation [9

9. M. Ley, J. Boudaden, and Z. T. Kuznicki, “Thermodynamic efficiency of an intermediate band photovoltaic cell with low threshold Auger generation,” J. Appl. Phys. 98(4), 044905 (2005). [CrossRef]

] etc. Most of the analysis put emphasis on how to quantify and theorize the quasi-Fermi level separation and how to apply the drift-diffusion model of current-voltage relationship. But we think it is more intuitive to parametrically analyze the partition of the involved carriers between different energy band transitions. We can assume certain percentage of carriers for the intermediate band absorption and others are for direct band gap absorption. By this way, we can avoid the tedious work of quasi-Fermi level calculation. Another often-overlooked facts are: it takes two photons for intermediate band absorption to generate an electron hole pair [2

2. A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]

]; and the photon flux density has to follow the general blackbody radiation rule and should be conserved. To include all these thoughts, we need to re-formulate the detailed balance model. In this work, we developed a concise, but realistic detailed balance model which should be useful for device designer and could shed some lights on the physical mechanism of IBSCs. At the end, we also apply our model and compare the degradation of open-circuit voltage (Voc) of different research groups. The comparison demonstrates the validity of our model and the potential of using this model for IBSC device analysis and design.

2. Intermediate band and photon quanta conservation

Figure 1
Fig. 1 Schematic diagram of an IBSC.
shows the schematic energy band diagram of an IBSC. The Eg1 is the largest energy band gap and usually is the band gap of the host semiconductor material. Eg2 and Eg3 represent the separation of intermediate band to valence band and conduction band, respectively, and they can be introduced by quantum dots, superlattices, or impurities [2

2. A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]

]. With the assumption of infinitesimal width of intermediate band, it is conceivable that Eg1 = Eg2 + Eg3. From Fig. 1, we can artificially assert that there are two portions of electrons/holes involved in the photovoltaic action: one is from Eg1 transition (denoted as “direct transition”), and the other is from Eg2 and Eg3, i.e. the intermediate band transition (IB). The combination of photo-generated carriers of these two transitions will flow into metal contact and become photovoltaic currents. The ratio or the partition between these two transitions (Direct vs. IB) was usually determined by quasi-Fermi level [2

2. A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]

]. In this paper, instead of calculating possible QFL separation or contemplating if QFL should be constant along the solar cell junction, we reverse this sequence and use parametric assignment of carrier partition as the inputs. The resulting calculation is based on how many carriers involve in the direct transition and the intermediate band transition respectively. We believe this should greatly simplify the calculation meanwhile it still provides useful information for the device designers.

There is one critical point, though, not very clearly clarify in the past. It is the conservation of the total number of incident photons from the sun, which is described best in a blackbody radiation form [2

2. A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]

] shown below:

N(ε1,ε2,T,μ)=2h3c2ε1ε2ε2dεe(εμ)/kT1,
(1)

2.1. General detailed balanced model

Ish=q×(photons from the sun-photons emitted by cell)q×(# of photons from the sun)=A×Eε2dεexp(ε/kTs)1,
(2)

where A is a constant combining electron charge, and the speed of light. The open-circuit voltage can also be calculated as [1

1. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]

]:

Voc=kTcqln(fEgε2dεexp(ε/kTs)1/Egε2dεexp(ε/kTc)1),
(3)

where f is a composite constant consisting of the ratio of radiative recombination rate to overall recombination rate, subtended angle, and the absorption probability of incident photons, etc. [1

1. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]

]. Tc and Ts are the temperatures of solar cell and the sun, respectively.

2.2. Assignment of percentage of involved carriers

Our photon-carrier relationship is in the reverse direction compared to the original IBSC theory: we assign certain percentages of carriers to direct transition and others are for indirect (intermediate band) transition. These photo-generated carriers then can be directly one-to-one mapped into photon distributions that belong to different transitions. Different energy bands can carry their own photon distributions. For transition of Egi (i = 1, 2, 3):

Ni(Egi,,Ts,0)=CiN(Egi,,Ts,0)=2Cih3c2Egiε2dεeε/kTs1,
(4)

where Ci (i = 1, 2, 3) is the corresponding portions of involved carriers for each band, and needs to be determined in our model.

2.3. Conservation of photons

Since the absorption events of a IBSC happen in such an infinitesimal time interval that the overall consumed photon population should be the same group in all three transitions, and thus it needs to be conserved. When the incident photon’s energy is higher than the direct transition (Eg1), all of them are absorbed and the total number of photon flux has to be the same as the general blackbody radiation formula (i.e. Eq. (1)); thus the following deduction must hold:

iNi(Eg1,,Ts,0)=N(Eg1,,Ts,0)[iCi]N(Eg1,,Ts,0)=N(Eg1,,Ts,0)iCi=1.
(5)

By comparing the two sides in Eq. (5), we find that the summation of Ci has to be one to satisfy the requirement of photon conservation at every wavelength.

2.4. Electronic isolation of IB

Since we assume the electronic isolation of IB, and we do not consider the re-emission or re-absorption issues, it is conceivable that the number of absorbed photons from valence band to IB shall equal to the number of elevated photons from IB to conduction band. So implementing this requirement, we have

C2N(Eg2,,Ts,0)=C3N(Eg3,,Ts,0).
(6)

This guideline is the same as the original IBSC theory [2

2. A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]

]. From Eqs. (5) and (6), when we parametrically assign a certain numerical number (<1) to one of the Ci coefficients, and the other two can be solved accordingly.

2.5. Modified efficiency of IB transition

Finally, the introduction of IB will cause the change of the original single band gap power conversion expression. Following the same notation in Shockley and Queisser paper [1

1. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]

], we can write down the corresponding ultimate efficiency as:

u(xg)=[C1xg1xg1x2dx(ex1)+C3xg3xg3x2dx(ex1)]/0x3dx(ex1).
(7)

3. Results and discussion

If we look into GaAs material system more closely, the calculation results in Fig. 3 agree with most of the experimental cases: the IBSC is worse than the single gap device. To probe further, we can analyze the device I-V characteristics. From published data, the change in Voc is usually responsible for the degradation of the PCE [3

3. T. Sugaya, S. Furue, H. Komaki, T. Amano, M. Mori, K. Komori, S. Niki, O. Numakami, and Y. Okano, “Highly stacked and well-aligned In0.4Ga0.6As quantum dot solar cells with In0.2Ga0.8As cap layer,” Appl. Phys. Lett. 97(18), 183104 (2010). [CrossRef]

5

5. R. Oshima, A. Takata, Y. Shoji, K. Akahane, and Y. Okada, “InAs/GaNAs strain-compensated quantum dots stacked up to 50 layers for use in high-efficiency solar cell,” Physica E 42(10), 2757–2760 (2010). [CrossRef]

,10

10. D. M. Chapin, C. S. Fuller, and G. L. Pearson, “A new silicon p-n junction photocell for converting solar radiation into electrical power,” J. Appl. Phys. 25(5), 676–677 (1954). [CrossRef]

12

12. S. Hubbard and R. Raffaelle, Boosting solar-cell efficiency with quantum-dot-based nanotechnology,” SPIE Newsroom, Feb. 8, 2010, http://spie.org/x39022.xml?highlight=x2358&ArticleID=x39022.

]. Using Eq. (3), we can approximately calculate the possible Voc in the proposed detailed balance model if ideal diode operation is assumed. The published works with EQE and Voc can be used for comparison to our model. The EQE is generally the quantum yield of generated electrons vs. incident photons. So the area underneath the EQE curves multiplied by the solar spectral photon flux density can be regarded as the number of carrier actually involved in the energy transition. Similar argument has been explored in [11

11. S. M. Hubbard, C. Plourde, Z. Bittner, C. G. Bailey, M. Harris, T. Bald, M. Bennett, D. V. Forbes, and R. Raffaelle, “InAs quantum dot enhancement of GaAs solar cells,” 2010 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010), pp. 1217–1222.

].

Figure 4
Fig. 4 Normalized Voc versus percentage of direct transition through Eg1, and the data points are from [5,11,12]. Calculation based on our model can be fitted well with different values of factor f.
shows our calculation against the measured Voc of GaAs based quantum dot solar cells [5

5. R. Oshima, A. Takata, Y. Shoji, K. Akahane, and Y. Okada, “InAs/GaNAs strain-compensated quantum dots stacked up to 50 layers for use in high-efficiency solar cell,” Physica E 42(10), 2757–2760 (2010). [CrossRef]

,11

11. S. M. Hubbard, C. Plourde, Z. Bittner, C. G. Bailey, M. Harris, T. Bald, M. Bennett, D. V. Forbes, and R. Raffaelle, “InAs quantum dot enhancement of GaAs solar cells,” 2010 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010), pp. 1217–1222.

,12

12. S. Hubbard and R. Raffaelle, Boosting solar-cell efficiency with quantum-dot-based nanotechnology,” SPIE Newsroom, Feb. 8, 2010, http://spie.org/x39022.xml?highlight=x2358&ArticleID=x39022.

]. The percentages of carriers for Eg1 transition can be obtained by comparing reference EQE to that of a QD device. By varying the f factor, we can see that the Voc data can be closely fitted to certain values of the f factor. Also the data points from the same group can be fitted with the same f factor (except for one data point, whose device performance is deteriorated due to emitter layer degradation [11

11. S. M. Hubbard, C. Plourde, Z. Bittner, C. G. Bailey, M. Harris, T. Bald, M. Bennett, D. V. Forbes, and R. Raffaelle, “InAs quantum dot enhancement of GaAs solar cells,” 2010 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010), pp. 1217–1222.

]). Since the f factor is a conglomerate of several coefficients [1

1. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]

], which differs from device to device, we can only conclude that it did distinguish between different groups. The main reason of Voc degradation can be the strong non-radiative recombination which often happens in quantum dot structures and thus lower the f factor significantly through the inclusion of the ratio between radiative and non-radiative recombination coefficients in its own definition. The other important factor we would like to point out is that the Voc degradation is unavoidable even from the most ideal detailed balance model with ideal diode characteristics [1

1. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]

]. The reason resides in Eq. (3), when the amount of available photo-generated carriers increases due to intermediate band absorption, the ratio of the integrals within the natural logarithm mathematically will drop. The resulting Voc/Vg will decrease accordingly.

This model can provide a good, but not precise estimate of the IBSC problem, since we did not consider other important factors in this model, such as voltage-dependent non-radiative recombination, re-emission and re-absorption between transitions, the non-standard ideality factor, saturation of quantum dots, etc. However, the trend is clear and the degradation of Voc can be improved by regulating the above-mentioned factors.

4. Conclusion

We developed a concise model with photon conservation criteria. The calculation revealed the enhancement of IB can be strong with wide band gap material such as GaN, and limited in smaller band gap material like GaAs. Meanwhile, it is also illustrative to use this model to calculate the degradation of open-circuit voltage, which can be a good tool to further improve the performance of IBSCs.

Acknowledgment

This work is supported under the projects of National Science Council, Taiwan, with grant numbers NSC 99-2221-E-009-052-MY3 and NSC100-3113-E-110-006.

References and links

1.

W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]

2.

A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]

3.

T. Sugaya, S. Furue, H. Komaki, T. Amano, M. Mori, K. Komori, S. Niki, O. Numakami, and Y. Okano, “Highly stacked and well-aligned In0.4Ga0.6As quantum dot solar cells with In0.2Ga0.8As cap layer,” Appl. Phys. Lett. 97(18), 183104 (2010). [CrossRef]

4.

T. Sugaya, S. Furue, O. Numakami, T. Amano, M. Mori, K. Komori, Y. Okano, and S. Niki, “Characteristics of highly stacked quantum dot solar cells fabricated by intermittent deposition of InGaAs,” in 2010 35th IEEE Photovoltaic Specialist Conference (PVSC) (IEEE, 2010), pp.1863–1867.

5.

R. Oshima, A. Takata, Y. Shoji, K. Akahane, and Y. Okada, “InAs/GaNAs strain-compensated quantum dots stacked up to 50 layers for use in high-efficiency solar cell,” Physica E 42(10), 2757–2760 (2010). [CrossRef]

6.

G. D. Wei, K.-T. Shiu, N. C. Giebink, and S. R. Forrest, “Thermodynamic limits of quantum photo-voltaic cell efficiency,” Appl. Phys. Lett. 91(22), 223507 (2007). [CrossRef]

7.

W. G. Hu, T. Inoue, O. Kojima, and T. Kita, “Effects of absorption coefficients and intermediate-band filling in InAs/GaAs quantum dot solar cells,” Appl. Phys. Lett. 97(19), 193106 (2010). [CrossRef]

8.

K. Yoshida, Y. Okada, and N. Sano, “Self-consistent simulation of intermediate band solar cells: Effect of occupation rates on device characteristics,” Appl. Phys. Lett. 97(13), 133503 (2010). [CrossRef]

9.

M. Ley, J. Boudaden, and Z. T. Kuznicki, “Thermodynamic efficiency of an intermediate band photovoltaic cell with low threshold Auger generation,” J. Appl. Phys. 98(4), 044905 (2005). [CrossRef]

10.

D. M. Chapin, C. S. Fuller, and G. L. Pearson, “A new silicon p-n junction photocell for converting solar radiation into electrical power,” J. Appl. Phys. 25(5), 676–677 (1954). [CrossRef]

11.

S. M. Hubbard, C. Plourde, Z. Bittner, C. G. Bailey, M. Harris, T. Bald, M. Bennett, D. V. Forbes, and R. Raffaelle, “InAs quantum dot enhancement of GaAs solar cells,” 2010 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010), pp. 1217–1222.

12.

S. Hubbard and R. Raffaelle, Boosting solar-cell efficiency with quantum-dot-based nanotechnology,” SPIE Newsroom, Feb. 8, 2010, http://spie.org/x39022.xml?highlight=x2358&ArticleID=x39022.

OCIS Codes
(040.5350) Detectors : Photovoltaic
(350.6050) Other areas of optics : Solar energy
(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

ToC Category:
Solar Energy

History
Original Manuscript: June 21, 2011
Revised Manuscript: July 28, 2011
Manuscript Accepted: August 8, 2011
Published: August 15, 2011

Citation
Chien-chung Lin, Wei-Ling Liu, and Ching-Yu Shih, "Detailed balance model for intermediate band solar cells with photon conservation," Opt. Express 19, 16927-16933 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16927


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. 32(3), 510–519 (1961). [CrossRef]
  2. A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. 78(26), 5014–5017 (1997). [CrossRef]
  3. T. Sugaya, S. Furue, H. Komaki, T. Amano, M. Mori, K. Komori, S. Niki, O. Numakami, and Y. Okano, “Highly stacked and well-aligned In0.4Ga0.6As quantum dot solar cells with In0.2Ga0.8As cap layer,” Appl. Phys. Lett. 97(18), 183104 (2010). [CrossRef]
  4. T. Sugaya, S. Furue, O. Numakami, T. Amano, M. Mori, K. Komori, Y. Okano, and S. Niki, “Characteristics of highly stacked quantum dot solar cells fabricated by intermittent deposition of InGaAs,” in 2010 35th IEEE Photovoltaic Specialist Conference (PVSC) (IEEE, 2010), pp.1863–1867.
  5. R. Oshima, A. Takata, Y. Shoji, K. Akahane, and Y. Okada, “InAs/GaNAs strain-compensated quantum dots stacked up to 50 layers for use in high-efficiency solar cell,” Physica E 42(10), 2757–2760 (2010). [CrossRef]
  6. G. D. Wei, K.-T. Shiu, N. C. Giebink, and S. R. Forrest, “Thermodynamic limits of quantum photo-voltaic cell efficiency,” Appl. Phys. Lett. 91(22), 223507 (2007). [CrossRef]
  7. W. G. Hu, T. Inoue, O. Kojima, and T. Kita, “Effects of absorption coefficients and intermediate-band filling in InAs/GaAs quantum dot solar cells,” Appl. Phys. Lett. 97(19), 193106 (2010). [CrossRef]
  8. K. Yoshida, Y. Okada, and N. Sano, “Self-consistent simulation of intermediate band solar cells: Effect of occupation rates on device characteristics,” Appl. Phys. Lett. 97(13), 133503 (2010). [CrossRef]
  9. M. Ley, J. Boudaden, and Z. T. Kuznicki, “Thermodynamic efficiency of an intermediate band photovoltaic cell with low threshold Auger generation,” J. Appl. Phys. 98(4), 044905 (2005). [CrossRef]
  10. D. M. Chapin, C. S. Fuller, and G. L. Pearson, “A new silicon p-n junction photocell for converting solar radiation into electrical power,” J. Appl. Phys. 25(5), 676–677 (1954). [CrossRef]
  11. S. M. Hubbard, C. Plourde, Z. Bittner, C. G. Bailey, M. Harris, T. Bald, M. Bennett, D. V. Forbes, and R. Raffaelle, “InAs quantum dot enhancement of GaAs solar cells,” 2010 35th IEEE Photovoltaic Specialists Conference (PVSC) (IEEE, 2010), pp. 1217–1222.
  12. S. Hubbard and R. Raffaelle, Boosting solar-cell efficiency with quantum-dot-based nanotechnology,” SPIE Newsroom, Feb. 8, 2010, http://spie.org/x39022.xml?highlight=x2358&ArticleID=x39022 .

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited