## Detailed balance model for intermediate band solar cells with photon conservation |

Optics Express, Vol. 19, Issue 18, pp. 16927-16933 (2011)

http://dx.doi.org/10.1364/OE.19.016927

Acrobat PDF (877 KB)

### 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

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 In_{0.4}Ga_{0.6}As quantum dot solar cells with In_{0.2}Ga_{0.8}As cap layer,” Appl. Phys. Lett. **97**(18), 183104 (2010). [CrossRef]

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]

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]

_{oc}) 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

_{g1}is the largest energy band gap and usually is the band gap of the host semiconductor material. E

_{g2}and E

_{g3}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]

_{g1}= E

_{g2}+ E

_{g3}. From Fig. 1, we can artificially assert that there are two portions of electrons/holes involved in the photovoltaic action: one is from E

_{g1}transition (denoted as “direct transition”), and the other is from E

_{g2}and E

_{g3}, 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

**78**(26), 5014–5017 (1997). [CrossRef]

**78**(26), 5014–5017 (1997). [CrossRef]

_{1}and ε

_{2}are the beginning and ending of photon energy, T is the temperature and μ is zero for photon distribution. In an ideal case, this photon flux N should be conserved at any time frame, and it puts an upper limit of the photon numbers that a solar cell can possibly receive. With this thought in mind, Fig. 2 illustrates the concept of our model and the comparison to the original one. The original IBSC concept [2

**78**(26), 5014–5017 (1997). [CrossRef]

_{g1}, between E

_{g2}and E

_{g1}, between E

_{g2}and E

_{g3}and below E

_{g3}(we assume E

_{g3}is smaller than E

_{g2}). Different mini-band transitions possess their own photon populations, which correspond to carriers involved in detailed balance model, as shown in the areas of A1~A3 of Fig. 2(c). In our model of IBSC, each transition can be treated as if they are in the single band gap situation, and the spectral line shapes in Fig. 2(c) are determined by the original blackbody radiation and the individual band gap. Each transition in this case, is proportionally scaled down by the criteria of photon conservation addressed in the next paragraphs. Between these transitions, the total number of photons involved should follow Eq. (1), since the number of photons at certain wavelength is not infinite. Therefore, we can set up general guidelines of our model as follows:

### 2.1. General detailed balanced model

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]

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]

_{c}and T

_{s}are the temperatures of solar cell and the sun, respectively.

### 2.2. Assignment of percentage of involved carriers

_{gi}(i = 1, 2, 3):

_{i}(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

_{g1}), 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:

_{i}has to be one to satisfy the requirement of photon conservation at every wavelength.

### 2.4. Electronic isolation of IB

**78**(26), 5014–5017 (1997). [CrossRef]

_{i}coefficients, and the other two can be solved accordingly.

### 2.5. Modified efficiency of IB transition

**32**(3), 510–519 (1961). [CrossRef]

_{gi}(i = 1, 2, 3) represents the different normalized energy transition in the IBSC, and it equals E

_{gi}/kT

_{s}(i = 1, 2, 3), T

_{s}is the temperature of the sun . Once these five guidelines are set, it is possible for us to solve C

_{i}, and calculate the PCE and open-circuit voltages under the constant photon flux assumption.

## 3. Results and discussion

_{g}) as two parameters. We can observe that IBSC design is not favorable in GaAs system, since the maximum efficiency happens at direct transition point. However, in the GaN system, the enhancement of IB is much more pronounced. The single band gap ultimate efficiency is 8.4%, but the introduction of intermediate band can boost the efficiency to 12.7%, which equals an increase of 1.5 times.

_{oc}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 In_{0.4}Ga_{0.6}As quantum dot solar cells with In_{0.2}Ga_{0.8}As cap layer,” Appl. Phys. Lett. **97**(18), 183104 (2010). [CrossRef]

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. 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. 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.

_{oc}in the proposed detailed balance model if ideal diode operation is assumed. The published works with EQE and V

_{oc}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].

_{oc}can be improved by regulating the above-mentioned factors.

## 4. Conclusion

## Acknowledgment

## References and links

1. | W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” J. Appl. Phys. |

2. | A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Phys. Rev. Lett. |

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 In |

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 |

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 |

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. |

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. |

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. |

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. |

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. |

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,” |

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

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### References

- 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]
- 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]
- 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]
- 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.
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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.
- 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 .

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