Subcontract Report

NREL/SR-520-47655

May 2010

Theoretical and Experimental 

Approaches to >50% Solar Cells 

December 14, 2004 – March 13, 2008

C. Honsberg

University of Delaware 

Newark, Delaware
National Renewable Energy Laboratory

1617 Cole Boulevard, Golden, Colorado 80401-3393

303-275-3000 • www.nrel.gov

NREL is a national laboratory of the U.S. Department of Energy

Office of Energy Efficiency and Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

Contract No. DE-AC36-08-GO28308

Subcontract Report

NREL/SR-520-47655

May 2010

Theoretical and Experimental 

Approaches to >50% Solar Cells 

December 14, 2004 – March 13, 2008

C. Honsberg

University of Delaware 

Newark, Delaware

NREL Technical Monitor: Fannie Eddy

Prepared under Subcontract No. XAT-5-44277-01
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1 

Summary

A central challenge in photovoltaics is to develop and implement approaches that allow 

photovoltaics to reach their thermodynamic efficiency limits. Tandem solar cells allow high 

efficiency, but as the number of solar cells in the tandem stack increases, they require 

increasing numbers of ideal, compatible materials and the incremental efficiency gains

become smaller. An alternative approach is to develop solar cell structures based on 

previously unutilized physical mechanisms. The central goal of the project is to develop a 

physical understanding, design rules, materials, and device approaches to implement 

advanced concept photovoltaics, focusing particularly on multiple quasi-Fermi level or 

intermediate-band approaches. An overview of the accomplishments is given below. 

• Demonstration that advanced concept approaches can be grouped in to one of five 

fundamental categories, and that the efficiency limits—and advantages and 

disadvantages— in each category are similar.

• Identification of fundamental loss mechanisms in multiple-energy-level solar cells. 

• Design rules for intermediate-band materials and calculation of efficiency contour 

plots for multiple-level solar cells, indicating how sensitive the efficiencies are to 

changes in band-structure parameters. 

• Thermodynamically and physically consistent model of the intermediate-band 

concept, including the effect of finite width of the band on the efficiency and the 

impact of absorption in the intermediate band, and calculation of absorption and 

emission rates. 

• Calculation of the impact of finite intermediate-band width and absorption within the 

intermediate-band efficiency.

• Calculation and inclusion of non-ideal absorption coefficients for an intermediatebandgap, spectral selectivity and impact of density of states on intermediate-band 

solar cell efficiency.

• Identification of materials used to implement multiple-energy-level solar cells which, 

based on ideal band structure, allow efficiencies of 50% at 1000x. 

• Calculation of ideal intermediate-band solar cell bandgaps under AM1.5 and finite 

concentration. 

• Epitaxial growth of the GaAsSb/GaAs system. 

• Epitaxial growth of InAs quantum dots on GaAsSb. 

• Observation of photoluminescence from the InAs quantum dots in GaAsSb. 

• Inclusion of strain in InAs/GaAsSb band-structure modeling. 

Thermodynamic grouping of advanced concept approaches. The large number of 

approaches and ways to implement advanced concept approaches can make comparison and 

analysis of solar cells based on such new concepts difficult. Despite the large number of 

approaches, from a thermodynamic standpoint, all advanced-concept approaches can be 

grouped according to which of the assumptions in thermodynamic modeling of conventional 

pn junctions it allows the solar cell to beneficially circumvent. The advantage of such a 
2 

grouping is that the design rules and losses are similar among approaches within a single 

class. The advanced-concept approaches and some implementations are shown in Table 1. 

Table 1: Approaches to solar cells that exceed Shockley-Queisser limit for a single solar cell.

Approach Advantages/Uses Central Issues Examples

Multiple 

spectrum

• Can be implemented using lowcost coatings

• Can use existing solar cells (or 

light-emitting diodes [LEDs] 

for thermophotonics)

Efficient conversion of 

solar spectrum not 

demonstrated

Thermophotovoltai

c 

Up and down 

conversion 

Multiple 

absorption 

• High impact-ionization rates 

demonstrated with colloidal 

quantum dots 

• Suited to conversion of highenergy photons

Transport of carriers 

not demonstrated

Multiple exciton 

generation

Two-photon 

absorption 

Raman absorption 

Multiple 

energy 

level

• Suited to low-energy photon 

conversion 

• Can capitalize on 

LED/photodetector devices

Need demonstration of 

multiple simultaneous 

radiative transitions of 

similar magnitude

Localized band 

(quantum well 

[QW]) 

Mini-band 

(intermediate-band 

solar cell)

Multiple 

temperate

Potential for high efficiencies 

using a single absorber material

Extraction of energy 

from hot-carrier 

populations not 

demonstrated 

Hot carriers 

QWs with thermal 

escape

AC solar 

cells

Potential for high efficiencies 

using a single absorber material

Requires THz devices Rectenna

Ideal band structure for intermediate-band solar cells

Figure 1

. The ideal intermediate-band 

structure has bandgap values of 1.95, 1.2, and 0.7 eV for maximum concentration under a 

blackbody spectrum. In addition to these values, there are several other requirements for the 

band structure of an intermediate-band solar cell as shown in a. These are that: (1) the 

valence-band offset is small; (2) there is only a single band or energy level between the 

conduction band and the energy levels formed by the overlap of quantum dots (QDs); and (3) 

the upper energy levels of the QD should not be within 3/2 kT of the continuum. 

Theoretically, the role of the valence band and conduction band may be switched, particularly

if the hole effective mass is smaller than the electron effective mass. Condition (1) arises from 

the fact that the hole effective masses are typically larger than electron effective masses, and 

if there is a valence-band offset, the energy levels in the valence band of the QD will be 

closely spaced, allowing thermalization between the energy levels and leading to the inability 

to maintain a separate quasi-Fermi level in the QD valence band compared to the barrier 

valence band. This, in turn, means that an offset in the valence band gives a loss in opencircuit voltage (VOC ). Similar arguments also give conditions (2) and (3). 
3 

 

Figure 1: (a) Schematic of band structure, showing ideal intermediate-band (in green) and non-ideal band 

structure elements (in red). (b) Band structure (neglecting strain) for InAs QD in GaAs barrier, showing 

presence of non-idealities. 

The additional requirements on band structure have a significant impact for materials for 

intermediate-band solar cells. For example, Figure 1b shows the band structure of an InAs 

QD (neglecting strain) with a GaAs barrier. Although the bandgaps are relatively close to 

their optimum values (and can be made ideal by replacing GaAs with AlGaAs), the band 

offsets and presence of additional bands means that such materials are not ideal. 

Design rules for intermediate-band materials and calculation of efficiency contour plots

for >60%. 

• Largest bandgap ECV ranges from 1.40 to 2.56 eV. 

The bandgaps and other band-structure features for an ideal material give rise to a 

limiting set of conditions that make it difficult to find an ideal material. However, by relaxing 

the constraint from an ideal material to one for about >50% at 1,000 suns (rather than 63% at 

maximum concentration), there is substantial relaxation in the allowable materials. The design 

rules for a QD intermediate-band (IB) material are: 

• Single, narrow IB with bandgap, ECI = 0.5 ECV - 0.2 eV. 

• Fermi energy, EF, located near to (within) the IB.

• Substrate commercially available with lattice constant between QD’s and barrier’s. 

• QD/barrier has minimum valence band offset. 

• Lattice mismatch > 1% between barrier and QD material.

Efficiency contour plots provide a graphical way of examining the relationship between 

bandgaps and efficiency. Figure 2 shows efficiency contour plots for 1,000x, showing that 

>55% efficiency can be realized with a range of bandgaps. 
4 

Figure 2: Efficiency contour plots for intermediate-band solar cell at 1,000x. 

Thermodynamic formulation for inclusion of non-idealities.

 

 The conventional 

thermodynamic formulation for an intermediate-band solar cell is poorly suited to inclusion of 

non-idealities. Three changes were implemented to extend the type and range of nonidealities. These include the following: (1) inclusion of finite width of intermediate band and 

non-idealities associated with this band; (2) new mathematical formulation for Bose-Einstein 

equation, allowing calculations with greater precisions and more rapid calculations; (3) 

inclusion of extended band-structure model, allowing calculation of absorption and emission 

rates, absolute rather than relative position of the Fermi-levels, and effects such as non-ideal 

absorption. 

Figure 3: (a) Schematic band structure for an intermediate-band solar cell, showing absorption for each 

band. (b) Schematic of the thermodynamic formulation for the intermediate-band solar cell, showing 

losses in the intermediate band and non-ideal absorption coefficients. 

Calculation of the impact of finite intermediate-band width and absorption within the 

intermediate band

Figure 4

. The impact of a finite width of the intermediate-band solar cell 

efficiency and optimum bandgaps is shown in and Figure 5. Figure 4 shows the 

impact assuming that there is no absorption in the intermediate band and demonstrating the 

necessity of the expanded thermodynamic formulation that takes into account absorption 

losses in the intermediate band. For example, as the width of the intermediate band increases, 

the efficiency is reduced to that approximately of a two-junction tandem. In the case of an 

“infinite” width intermediate band, this leads to the condition in which the material should act 

like a blackbody, but calculations show that it has substantial efficiency. If the absorption 

within the intermediate band is included as shown in Figure 5, then, as expected, the 

εCV=1.25e

εCV=3.13e
5 

efficiency drops to zero as the intermediate band width increases. However, it is notable that 

until the bandwidth is over 1 eV, the degradation in efficiency is relatively small.

(a) (b) 

Figure 4: (a) Efficiency as a function of intermediate band width for three different concentrations 

assuming that there is no absorption within the intermediate band. (b) Optimum values of the 

intermediate to conduction band energy and valence to intermediate band energy as a function of the 

intermediate band width.

 (a) (b) 

Figure 5: (a) Efficiency as a function of intermediate band width for three different concentrations 

assuming absorption within the intermediate band. (b) Optimum values of the intermediate to conduction 

band energy and valence to intermediate band energy as a function of the intermediate band width.

Calculation and inclusion of non-ideal absorption coefficients for an intermediate 

bandgap and impact of density of states.

Figure 6

A critical assumption in ideal intermediate band 

calculations is that of ideal absorption. Calculation of realistic absorption requires inclusion 

of an extended band diagram, including the density of states and transition matrix elements 

for the band structure. The assumed band structure is shown in a. The band structure, 

with additional assumptions about parameters such as the effective masses needed to calculate 

the transition probabilities, allows calculation of the absorption coefficients. The ideal 

absorption coefficients are shown in Figure 6b, and Figure 7 shows absorption coefficients 

calculated for a range of density of states. Figure 7 shows the importance of the density of 

states in realizing ideal intermediate band absorbers. 
6 

(a) (b) 

Figure 6: (a) Assumed band structure for an intermediate-band material, showing parabolic band 

assumption. (b) Ideal absorption coefficients for an intermediate-band material.

 (a) (b) 

 (c) (d) 

Figure 7: Calculated absorption coefficients for: the intermediate band (αII), the intermediate to 

conduction band (αCI); the valence band to intermediate band (αIV); and the valence to the conduction 

band (αCV). Plots (a) to (d) differ in the assumed density of states, increasing from (a) to (d). Absorption

coefficients are non-ideal in that they show: (a) overlap of absorption from different bands; (b) overlap of 

absorption from different bands; (c) incomplete absorption of the solar spectrum; and (d) incomplete 

absorption of the solar spectrum and overlap from different bands.

Candidate materials for intermediate-band solar cells

Table 2

. Based on the material design rules 

for intermediate-band solar cells determined above, a program to calculate the electronic 

states and band offsets for tertiary and binary combinations of III-V materials was used to 

determine candidate intermediate-band materials. To keep the computation time to a 

minimum and allow a search over the entire material space, effects such as strain are not 

included. The optimum material combinations are shown in . 
7 

Table 2: Potential material combinations for intermediate-band solar cells. The first and last rows have 

an efficiency of 48%, but are included as binary QDs, which are more practical than tertiary alloys. The 

other combinations have efficiency potential >50%.

Substrate Barrier QD ECV E∆CB

InP GaAs0.88Sb0.12 InAs 1.18 0.83 

InP GaAs InP0.85Sb0.15 1.42 0.49 

InP GaAs InAs0.40P0.60 1.42 0.49 

GaAs GaAs0.70P0.30 InP 1.79 0.43 

Calculation of bandgaps under AM1.5 and finite concentration.

Figure 8

A critical constraint in 

the material selection for intermediate-band solar cells is the combination of a large 

conduction-band offset and a small (near zero) valence-band offset. This constraint can be 

relaxed to some degree by considering different concentrations and spectra. As in 

conventional tandems, an AM1.5 spectrum reduces the maximum bandgaps due to the lower 

number of high-energy photons. The optimum bandgaps for the intermediate-band solar cell

is shown in for AM1.5 spectrum and 100x and 1,000x concentration. It shows that 

the difference between the global optimum intermediate bandgap and local optimum are 

relatively small, and further, that there is a larger range of optimum values. 

Figure 8: Optimum values of intermediate band to conduction band (ECI) and valence band to 

intermediate band (EIV) for AM.15 spectrum at (a) 100x and (b) 1,000x. 

Epitaxial growth of the GaAsSb/GaAs system and growth of InAs QDs on GaAsSb. The 

GaAsSb material system has multiple advantages for an intermediate-band material. The 

bandgap ranges are close to ideal, offering the potential for high performance. In addition, 

depending on the material composition, there is a Type I to Type II transition when grown on 

GaAs, thus allowing a material in which the valence-band offset is zero. Further, Sb is 

reported to act as a surface passivator for InAs QDs, improving the optical properties of the 

QDs. Finally, Sb has other advantageous properties on QD properties, increasing the areal 

density of InAs QDs and assisting in aligning of QD chains. Despite these advantages, the 

growth of GaAsSb requires care because—unlike III-III-V ternaries where the composition is 

determined by the flux ratios of the Group III elements over a range of temperatures—in a IIIV-V ternary, the composition also depends on the growth temperature. This becomes an issue 

a) b) 
8 

when growing QDs where the growth temperature of the barrier is typically different from

that for the QD growth. 

All of the samples studied were grown in an Applied Epi GenIII molecular beam epitaxy 

(MBE) system on semi-insulating GaAs (001) substrates. The Sb flux was provided by a 

Veeco all-PBN Corrosive Valved Cracker with the cracking zone held at 1000°C to ensure a 

Sb2-rich flux. A Veeco Valved Cracker was used to provide the As flux with the cracking 

zone held at 950°C to ensure an As2-rich flux. De-oxidation of the substrate surface under an 

As2 flux was observed at ~580°C as measured by an IRCON infrared pyrometer. 

Figure 9 summarizes the results. Two features become apparent regarding the dependence of 

the Sb content on the growth temperature and Sb flux. Firstly, as the temperature is increased, 

the Sb content decreases in a fairly linear manner in this range. This behavior fits with 

previous reports for high Sb content growth of GaAsSb where increased temperature saw the 

As content drop [1]. This behavior has been attributed to a As/Sb exchange reaction that 

increases with increased temperature [2]. Secondly, as expected, the Sb content increases 

with higher Sb flux, but it appears to be a non-linear dependence. This confirms previous 

results reported by Wu et al. [3], where the Sb was found to be very sensitive to changes in Sb 

flux for low Sb flux and content while being less sensitive for higher fluxes and contents. 

These results agree with predictions made by non-equilibrium thermodynamic analysis of IIIV-V compounds [4]. Further, as shown in Figure 9b, reciprocal space maps show that the 

defect density is low.

 (a) (b) 

Figure 9: (a) Sb content in GaAsSb as a function of the growth temperature and the Sb flux. The growth 

rate for all of the samples was ~0.40 monolayers/s. The dotted lines are guides to the eye only. (b) 

Reciprocal space maps obtained for two samples of ~12% Sb composition grown at T = 453°C.

Epitaxial growth of InAs quantum dots on GaAsSb.

Figure 10

 The addition of Sb changes the 

formation of InAs QDs compared to growing them on GaAs, both because Sb may act as a 

surfactant and also because the Sb composition changes the strain associated with the selfassembly of the QDs. shows an example of the QD grown on GaAs compared to 

GaAsSb, showing the improved performance of the QDs grown on GaAsSb. 

Reciprocal Space Map of 5 nm GaAsSb 
9 

 

 (a) (b) 

Figure 10: (a) InAs QD on GaAs; (b) InAs QDs on GaAs (5 monolayers) / GaAs1-xSbx (5nm) buffer layers 

with x =23%, with density 2.6 x 106 cm-2 

Measurement and modeling of photoluminescence from InAs QDs on GaAsSb.

Figure 11

 To be an 

effective intermediate-band material, the three energy levels in an intermediate-band solar cell 

must be radiatively coupled. Consequently, it is essential that the QD system has radiative 

emission. As a first step in demonstrating radiative emission between all three energy levels, 

there should be a radiative signature. Through collaboration with Prof. Stoleru in Material 

Science and Engineering at the University of Delaware, time-resolved photoluminescence 

was measured on InAs QD samples and samples without QDs. The results are shown in 

, with the expected band diagram shown in Figure 11b (not to scale). The results 

show a shift in the radiative signal with the InAs QDs, and results are consistent with the 

expected band diagram, showing a radiative transition of 1.1 eV and a calculated conductionband offset of about 300 mV. 

 

Figure 11: (a) Photoluminescence spectra of a InAs QD with a layer structure as shown in (b) and a 

GaAs/GaAsSb structure. 

Strain calculations for InAs QDs on GaAsSb material. To accurately determine the band 

structure of the QD material, strain must be incorporated into the band-structure calculations.

To calculate a more accurate band structure for InAs QDs on GaAsSb, k⋅p modeling for the 

250 nm 250 nm
10 

band structure was combined with strain models for pyramidal QDs. Figure 12 shows the 

strain calculations in a pyramidal QD. The impact of the strain on band structure was 

calculated using k⋅p modeling approaches. For an InAs QD on InPSb, the band structure is 

shown in Figure 13. For the compressive strain encountered in III-V materials, the 

conduction-band offset tends to decrease, degrading the efficiency potential of the 

intermediate-band approach. To compensate for this effect, the unstrained conduction-band 

offset should be increased; however, materials with such large conduction-band offsets and no 

valence-band offset are not common in the III-V material system. However, the expanded 

design space of the AM1.5G spectrum and variable concentration increases the flexibility in 

material choices, and the Sb-based material system with InAs QD allows relatively ideal 

intermediate-band materials, retaining over 90% of the theoretical maximum efficiency. 

Figure 12: Strain in a pyramidal InAs QD, showing the biaxial and hydrostatic strain in the QD.

Figure 13: Band structure for a pyramidal InAs QD on InPSb (a) neglecting strain and (b) including the 

strain profile. 

 

References

1. E. Selvig, B.O. Fimland, T. Skauli, and R. Haakenaasen, Journal of Crystal Growth 227-228, p562, 

2001.

2. Q. Xie, J.E. Van Nostrand, J.L. Brown, and C.E. Stutz, Journal of Applied Physics 86, p329, 1999.

3. S.D. Wu, L.W. Guo, W.X. Wang, Z.H. Li, X.Z. Shang, H.Y. Hu, Q. Huang, and J.M. Zhou, Journal of 

Crystal Growth 270, p359, 2004.

4. A. Yu. Egorov, A.R. Kvosh, V.M. Ustinov, A.E. Zhukov, P.S. Kop’ev, and C.W. Tu, Journal of 

Crystal Growth 188, p69, 1998.

-10 -5 0 5 10 15 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

z(nm)

Confining potential(eV)

-10 -5 0 5 10 15

0

0.5

1

1.5

2

z(nm)

Confining potential(eV)
F1146-E(10/2008) 

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4. TITLE AND SUBTITLE

Theoretical and Experimental Approaches To >50% Solar Cells

5a. CONTRACT NUMBER

DE-AC36-08-GO28308

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5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

C. Honsberg

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NREL/SR-520-47655

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PVA7.2001

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7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

University of Delaware

Electrical and Computer Engineering

Newark, DE 19716

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XAT-5-44277-01

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13. SUPPLEMENTARY NOTES

NREL Technical Monitor: Fannie Eddy

14. ABSTRACT (Maximum 200 Words)

A central challenge in photovoltaics is to develop and implement approaches that allow photovoltaics to reach their 

thermodynamic efficiency limits. Tandem solar cells allow high efficiency; but as the number of solar cells in the 

tandem stack increases, they require increasing numbers of ideal, compatible materials and the incremental 

efficiency gains become smaller. An alternative approach is to develop solar cell structures based on previously 

unutilized physical mechanisms. The central goal of the University of Delaware’s project is to develop the physical 

understanding, design rules, materials, and device approaches to implement advanced concept photovoltaics, 

focusing particularly on multiple quasi-Fermi level or intermediate-band approaches.

15. SUBJECT TERMS

PV; very high efficiency; thermodynamic efficiency limits; advanced concepts; multiple quasi-Fermi level approach; 

intermediate-band approach 

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OF ABSTRACT

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