Introduction

On the basis of solar cells incorporating quantum structures as wells (QWs) and dots (QDs), vast varieties of new concepts have been studied for improving conversion efficiency, such as increase in short-circuit current (Jsc) via excitonic absorption, multi-exciton generation, and multi-photon-absorption, and as increase in open-circuit voltage (Voc) via reducing mismatch between absorption and emission solid angles1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23. One of the most intensively studied is a type of QD solar cells with QDs embedded in a bulk host material, which are intended to realize the concept of intermediate-band (IB) solar cells1,2,3,4,5,6,7,8,9,10,11,12,13,14,15. Empirically, however, Jsc of such solar cells has only been moderately improved, while the open-circuit voltage (Voc) has been lowered, and the conversion efficiency of those cells has been lowered compared with bulk host-material solar cells4,5,6,7,8,9, which should be quantitatively analyzed in comparison with theories. In this work, we focus here on this issue.

This type of QD solar cells and their experimental data have mostly been compared with IB-solar-cell model theories1,2. In those theories, a chemical potential or carrier population in an IB or QD states is isolated from that in the host material due to strong phonon-bottleneck effects, and only the latter is connected to the external voltage, which is the key to implement the concept of IB solar cells with minor voltage degradation. Therein, carrier extraction is prohibited after single-photon absorption to QD states, but needs two- or more-photon absorption processes, that is, carriers that have been pumped into QD levels should be pumped again by absorbing a second photon into conduction band1,2. On the other hand, experimentally measured Voc and conversion efficiency of QD solar cells have been significantly lowered from bulk host-material solar cells in many cases4,5,6,7,8,9. The mechanism for these phenomena has to be investigated quantitatively or systematically. The growth of Stranski-Krastanov-mode self-assembled QDs may induce additional defects and/or dislocations due to strain accumulation, which may result in low material quality or radiative efficiency degrading Voc and conversion efficiency. Thus, the experimentally observed low Voc and conversion efficiency could be ascribed to the low material quality of QDs or host materials in QD solar cells. While researchers have continued their efforts to improve the growth of QDs8,9,10,11,12,13, improvements in Voc and conversion efficiency are still difficult to implement compared to the case of bulk-material cells without QDs. To escape this stalemate, quantitative examination on the basis of fundamental and general theories is necessary, to analyze whether the voltage drop and resulting reduced conversion efficiency originate from an intrinsic mechanism or from extrinsically inferior material quality.

Shockley-Queisser (S-Q) detailed-balance-limit theory is best suited for this purpose24,25. This theory has the excellent quality of allowing the determination of the upper limit of conversion efficiency with only the absorption spectrum of a solar cell, regardless of its structural details. More recently, we developed an extended theory to incorporate the extrinsic effects of non-ideal material quality, as indicated by the internal radiative efficiency (ηint) below unity26,27,28,29. The objective of this work is to use the extended detailed-balance-limit theory to quantitatively analyze the voltage drop and reduced conversion efficiency caused by intrinsic physics in QD solar cells, as apart from the contributions of extrinsically low material quality.

For QW solar cells, there has been an argument in the context of Shockley-Queisser detailed-balance theories that the conversion efficiency of QW solar cells cannot exceed that of a bulk cell with the optimum band gap, and that they are only useful in extending their band edges when no bulk material with the proper gap is found30,31. The same argument could be generally possible including the cases for QD and other quantum-structured solar cells, but no systematic or detailed study has been reported on this point.

In this work, we model the absorption spectrum of a QD solar cell as a simple two-step function at the host-absorption region (absorptivity is set as 1 for photons with energy greater than bulk-material bandgap Eg) and QD-absorption region (as a parameter a1 < 1 for photons with energy between Eg and the ground energy level arising from QD E1), demonstrated as Fig. 1 and formulated as Eq. (6), which is approximately comparable with reported experimental external quantum efficiency of QD solar cells8,15. Here all absorption processes are assumed as one-photon absorption, neglecting two- or multi-photon absorption processes. This model is used to evaluate the intrinsic and extrinsic upper limits of conversion efficiency with Jsc and Voc for quantum structural solar cells on the basis of the extended detailed balance theory within one-photon-absorption processes. Though we hereafter denote QD as a representative case, this absorption-spectrum model is very general and applicable not only to QDs, but also to other quantum or nano structures such as wells, wires, disks, and rods. The results clarify that the introduction of low-density QDs (QWs, wires, etc.) causes a significant drop in Voc with very a small gain in Jsc, hence resulting in very low efficiency. As the density (number) of QDs is increased, Jsc is increased proportionally, and the efficiency rises accordingly. When the density (number) of QDs becomes sufficiently high, the conversion efficiency, Jsc, and Voc become equal to those of a bulk solar cell made of the QDs material with a low bandgap. Note that these are intrinsic and unavoidable effects stemming from the absorption spectrum of the QDs solar cells. Extrinsic low material quality further degrades Voc and conversion efficiency.

Figure 1
figure 1

Modeled absorption spectrum of quantum-structural solar cells with a step-function tail below the host-material bandgap Eg.

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Results

We calculated the detailed-balance-limit value of conversion efficiency (ηsc) as a function of a1 (the absorptivity arising from QD) between 0 and 1, which is as explained in the method section28,29. Considering a typical example of InxGa1−xAs QDs embedded in a GaAs host material, we assume Eg = 1.4 eV and the binding energy (Eb = Eg − E1) to be between 0.001 eV and 0.6 eV. In addition to the radiative-limit case with internal radiative efficiencies of host/QD material (\({\eta }_{int}^{host/QD}=1\)), calculations were also performed for various other \({\eta }_{int}^{host}\) and \({\eta }_{int}^{QD}\), down to 10−5.

Figure 2 shows examples of the absorptivity spectra and calculated dark emission spectra of QD solar cells for various values of absorptivity a1 and binding energy Eb. In Fig. 2(a) with Eb = 0.05 eV, each dark emission spectrum clearly exhibits two peaks at Eg and E1, emitted from the host material and the QDs, respectively, whose intensities change with the values of α1L1. For large a1 (or large α1L1), the QD emission is dominant, whereas for moderate α1L1 around 0.1, both emissions from QD and host material are comparable, and for very small α1L1, the host material emission becomes dominant. For large binding energy Eb = 0.1 eV much greater than thermal energy ET = kBT ≈ 0.026 eV at temperature T = 300 K, emission mostly comes from QDs rather than host material, as shown in Fig. 2(b).

Figure 2
figure 2

Absorptivity spectrum (solid) and dark emission spectrum (dashed) of QD solar cells with a host material Eg of 1.4 eV, binding energies of (a) Eb = 0.05 eV and (b) Eb = 0.1 eV, and various QD absorptivities α1L1.

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Figure 3(a) shows the calculated detailed-balance-limit-conversion efficiency, ηsc, in the radiative limit (\({\eta }_{int}^{host/QD}=1\)). It converges to 33% at a1 = 0, corresponding to the efficiency limit of a bulk GaAs solar cell without QDs. It stays at 33% for various values of a1 if the QD binding energy, Eb, is smaller than 0.1 eV. However, in cases where Eb is above 0.1 eV, as a1 increases from 0, the conversion efficiency first drops drastically from a value of 33%, and then increases almost linearly. At a1 = 1, the conversion efficiency is nothing but that of a bulk solar cell with a bandgap of E1 = Eg − Eb. To clarify the origins of the behaviors of ηsc presented in Fig. 3(a), we next calculated the corresponding Jsc and Voc, as shown in Fig. 3(b,c).

Figure 3
figure 3

(a) Conversion efficiency, (b) Jsc, (c) Voc, and (d) effective bandgap of QD solar cells with host material Eg = 1.4 eV and ideal internal radiative efficiency \({\eta }_{int}^{host/QD}=1\) for varied a1 and Eb.

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In Fig. 3(b,c), Jsc and Voc are 32.7 mA/cm2 and 1.14 V, respectively, at a1 = 0 or when Eb < 0.1 eV, that is, the same values exhibited by a bulk GaAs solar cell with Eg = 1.4 eV and \({\eta }_{int}^{host/QD}=1\). At a1 = 1, they are identical to those of a bulk solar cell with a bandgap of E1. In Fig. 3(b), the boost of Jsc increases linearly as a1 increases, where the slopes increase as Eb increases, caused from the sub-band absorption add-on. For these cells only being introduced a few narrow-bandgap materials with very tiny a1, Jsc add-on is negligible, almost pinned at 32.7 mA/cm2, regardless of how much Eb is. On the other hand, as a1 increases in Fig. 3(c), Voc drops very drastically and steeply near a1 = 0, flattens at a1 below 0.1, and then converges to the values at a1 = 1. It is evident that the conversion efficiency is almost proportional to the product of Jsc and Voc. In fact, we confirmed that fill factors do not change much in the present parameter regions24.

The significant inherent drop in Voc near a1 = 0 in Fig. 3(c) can be interpreted, via expression of Voc;

$$\begin{array}{rcl}{V}_{oc}({a}_{1},{E}_{1},{E}_{g}) & = & {V}_{T}\,\mathrm{ln}({J}_{sc}/{J}_{0})\\ & = & {V}_{T}\,\mathrm{ln}\,{J}_{sc}-{V}_{T}\,\mathrm{ln}\,q({R}_{ext0}^{host}+{R}_{ext0}^{QD})\\ & = & {V}_{T}\,\mathrm{ln}\,{J}_{sc}-{V}_{T}\,\mathrm{ln}\,q{R}_{ext0}^{host}-{V}_{T}\,\mathrm{ln}(1+\frac{{R}_{ext0}^{QD}}{{R}_{ext0}^{host}})\end{array}$$
(1)

derived from Eqs (8–11) with \({\eta }_{int}^{host/QD}=1\), where VT = kBT/q ≈ 0.026 V is the thermal voltage, and \({R}_{ext0}^{host}\) and \({R}_{ext0}^{QD}\) are the radiative recombination flux from host material and QD under dark condition, at T = 300 K. Here, dependence of Voc on a1 via Jsc is small, and that via the changes of the dark current (J0) is dominant. Then, for \({E}_{b}\gg {E}_{T}\)(=VTq), Eq. (1) becomes approximately as

$${V}_{oc}({a}_{1},{E}_{1},{E}_{g})={V}_{oc}^{Bulk}({E}_{g})-{V}_{T}\,\mathrm{ln}(1+{a}_{1}\frac{{E}_{1}^{2}}{{E}_{g}^{2}}\,\exp \,\frac{{E}_{b}}{{E}_{T}})$$
(2)

At a1 = 0, Voc is the value of open-circuit voltage \({V}_{oc}^{Bulk}({E}_{g})\) for host-material-bulk cells with Eg. As a1 is increased from 0, Voc goes down steeply, because coefficient exp(Eb/ET) of a1 in the second term in Eq. (2) is very large, because of \({E}_{b}\gg {E}_{T}\).

For \(\exp ({E}_{b}/{E}_{T})\gg 1/{a}_{1}\), Eq. (2) is expressed approximately as,

$${V}_{oc}({a}_{1},{E}_{1},{E}_{g})={V}_{oc}^{Bulk}({E}_{1})-{V}_{T}\,\mathrm{ln}\,{a}_{1}$$
(3)

Here, the contribution of VT ln a1 is smaller than 60 mV for 0.1 < a1 < 1, and thus Voc is close to open-circuit voltage \({V}_{oc}^{Bulk}({E}_{1})\) for bulk cells with E1. This arises from that J0 in cells with low E1 for large Eb is almost governed by the recombination current arising from QD \((q{R}_{ext}^{QD})\), with band edge at E1 and absorptivity a1. Crudely speaking, the more a1 and the lower E1 indicate the more radiative emission losses from QD and further more dark current, which lowered Voc.

Figure 3(d) plots the emission energy, Eem, defined as the center-of-mass energy in the emission spectra. For a large binding energy of Eb > 0.1 eV, Eem immediately drops to E1 as a1 increases from 0. For shallow bonding energy Eb < 0.1 eV, Eem drops only slightly and more gradually and stays close to the host bandgap, Eg. The behaviors of Eem represent the changes in their emission spectra in Fig. 2(a,b), where the position of the dominant emission peak is switched from Eg to E1 as a1 and Eb increases. In the detailed-balance-limit theory with a radiative limit (\({\eta }_{int}^{host/QD}=1\)), carrier loss only occurs via radiative emission, which is determined by the product of absorptivity a(Eem) and the 300-K blackbody emission intensity, B(Eem), at the emission energy Eem. Therefore, Eem can be interpreted as the effective band-gap energy determining Voc, corresponding to the first term in Eq. (3) mentioned above. This explains why Voc drops steeply as a1 increases from 0, and its feature versus E1 at moderate and large a1 in Fig. 3(c) are similar to those of Eem in Fig. 3(d).

Note that the results in Figs 2 and 3 are all obtained in the radiative limit with \({\eta }_{int}^{host/QD}=1\). Therefore, these significant drops in the open-circuit voltage and conversion efficiency are intrinsic and unavoidable consequence of the absorptivity spectra of QD solar cells modeled in Fig. 1.

Figure 4 exhibits the conversion efficiency (a, c, e), Jsc and Voc values (b, d, f) of QD solar cells with varied material qualities arising from extrinsic origins or internal radiative efficiencies, ηint, below 1 down to 10−5. Here, we assumed QD and host materials have the same ηint. Three typical values are assumed for the binding energy Eb, namely 0.01 eV (a, b), 0.3 eV (c, d), and 0.6 eV (e, f). A black curve in each panel represents data in the radiative limit (ηint = 1) without non-radiative recombination losses. We note that a slight drop of ηint from 1 to 0.9 already causes a drastic downward shift in conversion efficiency by about 2~3% absolute and in Voc by 0.05~0.1 V. Significant drops in conversion efficiency and Voc also occur as ηint degrades from 1 to 0.1, in this case by about 5~10% absolute and about 0.2 V, respectively. Further drops in conversion efficiency and Voc with degradation of ηint by two orders of magnitude from 0.1 (or 10−3) to 10−3 (or 10−5) are by about 4~6% absolute and about 0.1~0.15 V, respectively. In addition, extrinsic drops slightly increase as a1 increases. Especially at a1 close to 1, conversion efficiency and Voc for non-unity ηint steeply drop to the corresponding values for bulk cells with bandgap E1, notably differing from the gradual black curves of ηint = 1. This arises from a sharply increasing non-radiative recombination losses in cells with non-unity ηint, when a1 approaching to 1. In this case, α1L1 becomes extremely large, which makes ηext sharply reduced in Eq. (12) for ηint < 1, and causes the drops of Voc and conversion efficiency.

Figure 4
figure 4

(a,c,e) Conversion efficiency, (b,d,f) Jsc (dashed) and Voc (solid) of QD solar cells with host material Eg = 1.4 eV for varied a1 and \({\eta }_{int}^{host/QD}\) at Eb = 0.01 eV, 0.3 eV, and 0.6 eV.

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Figure 5 exhibit the cell behaviors whose QD and host material qualities are different: three typical values are assumed for \({\eta }_{int}^{QD}\), namely 1 (a, b), 0.1 (c, d), and 0.0001 (e, f), and in each subplot \({\eta }_{int}^{host}\) are set as 1, 0.9, 0.1 and Eb also taken by 0.01, 0.3, 0.6 eV, respectively. Although the drop tendencies are similar to those with the same Eb in Fig. 4, they also reveal some different behaviors. For large Eb of 0.3 and 0.6 eV, Voc and conversion efficiency for different \({\eta }_{int}^{host}\) are almost overlapping and only determined by a1, E1 and \({\eta }_{int}^{QD}\), because the behaviors of such cells with deep Eb greater than several ET are very similar as those of bulk cells with bandgap E1. These are reasonable, because Voc for large Eb is approximately expressed as

$$\begin{array}{rcl}{V}_{oc} & \approx & {V}_{oc}^{radiative}+{V}_{T}\,\mathrm{ln}\,{\eta }_{ext}^{QD}\\ & = & {V}_{oc}^{radiative}-{V}_{T}\,\mathrm{ln}\,[1+\frac{4{n}^{2}{\alpha }_{1}{L}_{1}}{{a}_{1}}(\frac{1}{{\eta }_{{int}}^{QD}}-1)],\end{array}$$
(4)

which is independent of the host material quality \({\eta }_{int}^{host}\). Note that the 1st and 2nd terms in Eq. (4) serve as intrinsic and extrinsic drop, respectively.

Figure 5
figure 5

(a,c,e) Conversion efficiency, (b,d,f) and Voc of QD solar cells with host material Eg = 1.4 eV for varied a1 and \({\eta }_{int}^{QD}\) at Eb = 0.01 eV, 0.3 eV, and 0.6 eV and \({\eta }_{int}^{host}=1\), 0.9, and 0.1.

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In contrast, for cells with shallow Eb, Voc is determined both of host and narrow-gap materials. For low a1, where recombination current in host band dominates the dark current, the conversion efficiency and Voc almost converge to the values of bulk cells with Eg and \({\eta }_{int}^{host}\), and are weakly dependent on a1 and \({\eta }_{int}^{QD}\). However, in the large a1 region where the recombination in QD becomes dominant, Voc drops primarily by E1 and \({\eta }_{int}^{QD}\). Note here these colored curves in Figs 4 and 5 include both intrinsic and extrinsic drops in conversion efficiency and Voc, which are comparable with realistic data.

Discussion

The above essential features of intrinsic and extrinsic drops in conversion efficiency and Voc hold not only for the simple two-step-function model, but also for a finite-band-width model, where absorption band of QDs with absorptivity a1 starts at E1 and ends at E2(<Eg) so that absorptivity gap exists between E2 and Eg15,30,32. For example, we analyzed a case with E2 = E1 + 3ET, and found that Voc stays almost the same as Fig. 3(c) obtained for the simple two-step-function absorptivity model, though Jsc is decreased due to the absorptivity gap between E2 and Eg, and conversion efficiency is lowered accordingly. In short, the intrinsic Voc drops are not sensitive to sub-band absorption profiles, but are sensitive to the energy position of the lowest absorption edge and the absorptivity amplitude at the edge. Indeed, this conclusion is also consistent with reports on the detailed-balance-limit efficiency for semiconductors with inhomogeneous alloy broadening with Gaussian tails33,34,35 or with excitonic band-edge peaks32. It is of course consistent with previous reports on the detailed-balance-limit efficiency of QW solar cells30. It is interesting to examine our presented conversion-efficiency results at various a1 and Eb with the well-known ultimate efficiency as in S-Q paper24, that is, detailed-balance limit for the same two-step-function-absorption cells with temperature of 0 K under 6000 K-blackbody sun, expressed as,

$$\begin{array}{c}u({x}_{1},{x}_{g},{a}_{1})=\{\begin{array}{cc}\frac{{x}_{g}\,{\int }_{{x}_{g}}^{{\rm{\infty }}}\,{x}^{2}dx/({e}^{x}-1)}{{\int }_{0}^{{\rm{\infty }}}\,{x}^{3}dx/({e}^{x}-1)}, & {\rm{i}}{\rm{f}}\,{a}_{1}=0\,\\ \frac{{a}_{1}{x}_{1}\,{\int }_{{x}_{1}}^{{x}_{g}}\,\frac{{x}^{2}dx}{{e}^{x}-1}+{x}_{1}\,{\int }_{{x}_{g}}^{{\rm{\infty }}}\,\frac{{x}^{2}dx}{{e}^{x}-1}}{{\int }_{0}^{{\rm{\infty }}}\,{x}^{3}dx/({e}^{x}-1)}. & {\rm{i}}{\rm{f}}\,{a}_{1}\ne 0\end{array}\end{array}$$
(5)

We plot, in Fig. 6(a), the ultimate efficiency by dashed lines, in comparison with our results in Fig. 3(a), for the three cases of Eb = 0.01, 0.3, and 0.6 eV (E1 = Eg − Eb = 1.39 eV, 1.1 eV, and 0.8 eV). We marked the two-limit points of a1 = 0 (filled symbols) and a1 = 1 (open symbols) for the ultimate efficiency (squares) and our present results (circles), which should be equal to the results of S-Q paper24. Figure 6(b) shows the ultimate efficiency and S-Q-limit efficiency for conventional single-step-function absorption model24, and indeed, the marked data points at Eg = 1.4 eV, 1.39 eV, 1.1 eV, and 0.8 eV are consistent with those marked by the corresponding symbols in Fig. 6(a).

Figure 6
figure 6

Comparison of ultimate efficiency of QD solar cells at 0 K with the efficiency of 300 K cells under AM1.5 for varied a1 and Eb = 0.01 eV, 0.3 eV, and 0.6 eV.

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Note in Fig. 6(a), that the intrinsic drop also occurs in ultimate efficiency at 0 < a1 < 1 brought by the narrow-gap materials introduction, similarly to our present results of QD cells at 300 K. At 0 K, all carriers relax down to the lowest possible energy levels of E1 and the recombination from QD confined level is the only recombination mechanism, which cause Voc drop in the ultimate efficiency limit going down to E1/q. However, at a finite temperature T, typically for 300 K assumed in this work, all carriers generated from absorbed photons reach a Boltzmann distribution at T, and, as a result, Voc drops down to below E1/q, resulting in our present efficiency limit curves being softer than those of ultimate limit.

It is important to emphasize that the detailed-balance-limit study with the two-step-function model is applicable not only for QW solar cells, but also for QD and other quantum-structure solar cells. Whether or not theoretical models and assumptions are correct for each case of experiments should be judged by examining agreements between experimental and theoretical results. Our present work provides systematic theoretical results for all of conversion efficiency, Voc, Jsc, and emission photon energy Eem for such comparison.

One of the major and long lasting questions in solar-cell study is how close practical QD solar cells with embedded QDs in host materials are to the concept of IB solar cells3,4,8,13. The IB-solar-cell theoretical model1,2 assumes the phonon bottleneck effect, which was theoretically predicted for QDs having a discrete energy levels36,37: As the level spacings in QDs become larger, calculated carrier relaxation rates in QDs mediated by single phonon emission become slower due to energy and momentum conservation. This effect has been controversial, and consensus has not been established yet. In the presence of the phonon bottleneck effect, populations, and hence chemical potentials, of electrons and holes in the host material are isolated from those in an IB or QD states. Therefore, incorporation of IB or QD states in the middle of a pn-junction should cause only a small intrinsic Voc drop. Moreover, single photon absorption in QDs cannot contribute to photo-current, but two- or multi-photon absorption processes are necessary to generate photo-current. The concept of IB solar cells is based on these grounds. On the other hand, our present detailed-balance calculations assume that the phonon bottleneck effect is absent or negligible. Then, the thermalization of conduction-band electrons across the host material and QDs is much faster than electron-hole recombination. In other words, the chemical potentials of conduction-band electrons are equal between in the host material and in the QDs. The same things are true for valence-band holes. Therefore, the intrinsic drop in Voc occurs, and photo-current is induced via one-photon absorption processes to both of QDs and the host material.

Note that the IB-solar-cell model and our present detailed-balance model respectively deal with the opposite limits of slow (no) versus fast (instantaneous) carrier thermalization across the host material and QDs, assuming existence versus absence of the phonon bottleneck effect for carrier relaxation in QDs. It should be important to compare experimental data of QD solar cells with calculations via various possible models in the equal footing.

We point out that experimental characteristics of QD solar cells reported so far mostly showing degraded Voc from those of reference-bulk solar cells4,5,6,7,8,9 may be explained as intrinsic by our present calculations, if fast carrier thermalization across the host material and QDs is the case. We note that this can be checked via electroluminescence or photoluminescence experiments by excitation into the host materials, because it predicts whether or not the luminescence intensities of QDs and a host-material follow the same Boltzmann distribution with the identical chemical potential and temperature, as shown in Fig. 2.

The efficiency calculated here on the basis of detailed-balance-limit theory under single-photon absorption only showed the case where embedded QDs bring down the performance of the host-material bulk cells. However, as we mentioned at the beginning of the introduction section, vast varieties of new concepts have been proposed for QD solar cells, and significant increase in short-circuit current, for example, via multi-photon absorption and multi-exciton generation, may compensate the voltage drops and result in overall improvement in conversion efficiency. Our present calculation results should be very important to evaluate such a break-even point for the new-concept QD solar cells, or to quantitatively analyze extrinsic and intrinsic drops in experimental conversion efficiency of fabricated samples of QD solar cells. In this sense, this work should be practically helpful in developing all of the new-concept QD solar cells.

Conclusion

In summary, we have presented an analysis of the detailed-balance-limit conversion efficiency, short-circuit current, open-circuit voltage, and emission energy of QD solar cells with various parameters for the QD-absorption band below the host-material band gap. When the QD-absorption band has absorptivity, a1, of almost 0 or small QD-binding energy Eb below 0.1 eV, the cell is almost identical to a bulk-host-material solar cell. As a1 increases from 0 with Eb > 0.1 eV, Jsc increases linearly while Voc drops steeply near a1 = 0 and becomes flat for a1 > 0.1. Nearly proportionally to the product of Jsc and Voc, the conversion efficiency drops sharply near a1 = 0 and then increases almost linearly. The center-of-mass emission energy, Eem, plays the role of effective band-gap energy to determine Voc. Additional drops in conversion efficiency and Voc occur with extrinsic degradation of material quality or internal radiative efficiency, ηint. Our results suggest that drops of Voc and conversion efficiency in QD solar cells may be caused by these intrinsic reasons as a result of fast carrier thermalization across the host material and QDs.

Methods

Figure 1 shows the simple two-step-function absorptivity (a), related to the product of absorption coefficient (α) and material thickness (L), as a = 1 − exp(−2αL), which are given respectively by

$$\begin{array}{c}a=\{\begin{array}{cc}1, & {\rm{i}}{\rm{f}}\,E\geqslant {E}_{g}\\ {a}_{1}, & {\rm{i}}{\rm{f}}\,{E}_{g} > E\geqslant {E}_{1}\\ 0, & {\rm{i}}{\rm{f}}\,E < {E}_{1}\end{array}\end{array}$$
(6)

and

$$\begin{array}{c}\alpha L=\{\begin{array}{cc}5, & {\rm{i}}{\rm{f}}\,E\geqslant {E}_{g}\\ {\alpha }_{1}{L}_{1}, & {\rm{i}}{\rm{f}}\,{E}_{g} > E\geqslant {E}_{1}\\ 0, & {\rm{i}}{\rm{f}}\,E < {E}_{1}\end{array}\end{array}$$
(7)

for the absorption spectrum of a QD solar cell. Emphasize again, though we denote QD as a representative case in this paper, this absorption-spectrum model and conclusion are applicable not only to QDs, but also to other quantum or nano structures. Also all absorption processes are considered as one-photon absorption, rather than the two- or multi-photon absorption. In Eqs (6 and 7), the following assumptions are also made: the host material with bandgap Eg is thick enough to have an above-Eg absorptivity of nearly unity, while QDs with binding energies of Eb extend the low-energy absorption-band edge to E1 = Eg − Eb with an absorptivity of a1. Effects of more limited absorption band width of QDs are discussed in the discussion part of this paper. To investigate the upper-limit efficiency, it is also assumed that the solar cell has a perfect rear mirror to enable double-pass absorption. αL is taken to be greater than 5 above Eg, while to be various values represented by parameter α1L1 for energies between Eg and E1. The density and absorption-oscillator strength of the QDs determine α1L1.

Once the absorptivity spectrum a(E) is given, the Kirchhoff law of radiation or the detailed-balance principle between photon absorption and emission with the Planck’s radiation formula for 300-K blackbody emission provides the dark emission spectrum of the solar cell at 300 K25. Under a Boltzmann-statistics approximation, the emission spectrum under bias voltage V is equal to the product of the dark emission spectrum and a voltage factor of exp(V/VT), where VT = kBT/q ≈ 0.026 V is the thermal voltage at T = 300 K. Here we make an implicit assumption that photo-generated electrons and holes are in respective thermal equilibrium with the same carrier temperature of 300 K but with separated respective chemical potentials. Additionally, infinite carrier mobility, such that the difference in the chemical potentials of electrons and holes is uniform over the p-n junction and equal to the product of bias voltage and electron charge qV, is assumed in this model. This assumption is not realistic, but ideal, whose use is justified when evaluating the ideal theoretical upper limit of conversion efficiency. The Boltzmann-statistics approximation is known to cause deviation from rigorous Fermi-Dirac statistics for strongly concentrated illuminations, deep QDs, or strongly doped QDs, so we checked that the deviation is negligibly small for undoped QDs with the parameter regions covered in this paper.

By the carrier balance in a solar cell, the current J flowing out from the cell is equal to the difference between the photocurrent Jsc, and the recombination-loss current arising from host and QD materials. Thus, the I-V characteristics are given by

$$J={J}_{sc}-q(\frac{{R}_{ext}^{host}}{{\eta }_{ext}^{host}}+\frac{{R}_{ext}^{QD}}{{\eta }_{ext}^{QD}})={J}_{sc}-q(\frac{{R}_{ext0}^{host}\,\exp \,\frac{V}{{V}_{T}}}{{\eta }_{ext}^{host}}+\frac{{R}_{ext0}^{QD}\,\exp \,\frac{V}{{V}_{T}}}{{\eta }_{ext}^{QD}})$$
(8)
$${J}_{sc}=q\,{\int }_{0}^{\infty }\,a(E)S(E)dE$$
(9)
$${R}_{ext0}^{host}=\pi \,{\int }_{{E}_{g}}^{\infty }\,a(E)B(E)dE$$
(10)
$${R}_{ext0}^{QD}=\pi \,{\int }_{{E}_{1}}^{{E}_{g}}\,a(E)B(E)dE$$
(11)

where S(E) and B(E) are solar and 300-K blackbody spectra24, respectively. In this paper, we use the solar spectrum of AM1.5 G with an incident power per unit area of Pin = 100 mW/cm2, and take only one-photon-absorption processes into account. The current-loss term \(q{R}_{ext}^{host}/{\eta }_{ext}^{host}+q{R}_{ext}^{QD}/{\eta }_{ext}^{QD}\) in Eq. (8) includes radiative-recombination current loss from host- and QD-materials for external emission loss via front surface (\(q{R}_{ext}^{host}+q{R}_{ext}^{QD}\)) and non-radiative-recombination current loss (indicted by \({\eta }_{ext}^{host}\) and \({\eta }_{ext}^{QD}\)). \(q{R}_{ext0}^{host}\) and \(q{R}_{ext0}^{QD}\) respectively represent the radiative-recombination current loss via the front surface in the dark. We apply the relation between external and internal radiative efficiency (ηext and ηint)28,29, as,

$$\frac{1}{{\eta }_{ext}^{host/QD}}-1=\frac{4{n}^{2}\overline{{\alpha }^{host/QD}}L}{\overline{{a}^{host/QD}}}(\frac{1}{{\eta }_{{int}}^{host/QD}}-1)$$
(12)
$$\overline{{a}^{host}}={\int }_{{E}_{g}}^{\infty }\,a(E)B(E)dE/{\int }_{{E}_{g}}^{\infty }\,B(E)dE$$
(13)
$$\overline{{a}^{QD}}={\int }_{{E}_{1}}^{{E}_{g}}\,a(E)B(E)dE/{\int }_{{E}_{1}}^{{E}_{g}}\,B(E)dE$$
(14)
$$\overline{{\alpha }^{host}}={\int }_{{E}_{g}}^{\infty }\,\alpha (E)B(E)dE/{\int }_{{E}_{g}}^{\infty }\,B(E)dE$$
(15)
$$\overline{{\alpha }^{QD}}={\int }_{{E}_{1}}^{{E}_{g}}\,\alpha (E)B(E)dE/{\int }_{{E}_{1}}^{{E}_{g}}\,B(E)dE$$
(16)

to introduce internal radiative efficiency \(({\eta }_{int}^{host/QD})\) as pure indicators of the host/QD material quality of single-junction QD solar cells, separately from the cell geometry. Here, n and L are the reflective index and material thickness. \(\bar{<mml:mpadded xmlns:xlink="http://www.w3.org/1999/xlink" lspace="-1.5pt">{a}^{host/QD}</mml:mpadded>}\) and \(\bar{<mml:mpadded xmlns:xlink="http://www.w3.org/1999/xlink" lspace="-1.5pt">{a}^{host/QD}</mml:mpadded>}\) are the corresponding energy-averaged absorptivity and absorption coefficient for host/QD materials, respectively.