Increasing conversion efficiency of two-step photon up-conversion solar cell with a voltage booster hetero-interface

Development of high-efficiency solar cells is one of the attractive challenges in renewable energy technologies. Photon up-conversion can reduce the transmission loss and is one of the promising concepts which improve conversion efficiency. Here we present an analysis of the conversion efficiency, which can be increased by up-conversion in a single-junction solar cell with a hetero-interface that boosts the output voltage. We confirm that an increase in the quasi-Fermi gap and substantial photocurrent generation result in a high conversion efficiency.

transition in the type II quantum confined system, is spatially indirect, and, thereby, the interband-oscillator strength is lower than that of the type I quantum confined system. In addition, excited electrons in the CB quickly relax to the IB 18 . According to the theoretical simulation performed by Tomić, non-radiative relaxation time of electrons in the CB into the IB is few picoseconds for the quantum-dot IBSCs 19 . This fast process is unavoidable and also gives rise to reduction of the conversion efficiency.
If we can spatially draw the IB out of the bandgap, electrons excited in the CB will be prevented from relaxing to the IB. Recently, we proposed a two-step photon up-conversion (TPU)-SC 20 realising the ratcheting process [21][22][23] using a hetero-interface comprising different semiconductor bandgaps. In the case of a TPU-SC fabricated on a p-type substrate, below-gap photons passing through the wide bandgap semiconductor (WGS) layer excite the narrow bandgap semiconductor (NGS), in which excited electrons drift towards the WGS/NGS hetero-interface and accumulate there, while holes reach the p-layer. The long-lived electron, which was prevented from recombining at the hetero-interface is then efficiently excited by another below-gap photon in the NGS and lifted to the CB of the WGS. This TPU boosts the output voltage at the hetero-interface. In the TPU-SC, the two-step excitation in our device does not produce a high-energy photon but creates a free carrier in the CB. The potential energy of the CB is far from each excitation photon energy. Using two photons, an electron is finally pumped from the VB to the CB by way of the CB of the NGS. Here, the carrier energy is up-converted by two photons, and the excitation from the CB to the VB is equivalent to an event caused by a high-energy photon.
In this work, we study the increase in conversion efficiency of a TPU-SC with a voltage booster hetero-interface in the detailed-balance framework, considering a steady state between the carrier generation and recombination at the optimum operation point of an SC. We clarify the band discontinuity effects on the conversion efficiency between the NGS and WGS and offer a route to high-conversion-efficiency solar cells exceeding 50%. Figure 1 shows a schematic band diagram of a typical TPU-SC. The TPU-SC is composed of a single-diode structure containing a hetero-interface which consists of a WGS and NGS. It is noted that the IB in this TPU-SC is the CB of the NGS. Sunlight irradiates the WGS side (left-hand side of Fig. 1), high-energy photons are absorbed in the WGS layer, and excited electrons and holes drift in opposite directions towards the n-and p-type electrodes, respectively. In the case of a TPU-SC fabricated on a p-type substrate, below-gap photons passing through the WGS layer excite the NGS, in which excited electrons drift towards the WGS/NGS hetero-interface and accumulate there, while holes reach the p-layer. The spatial carrier separation prevents the recombination of electrons with holes and extends the electron lifetime. Electrons with longer lifetimes have a greater potential for intraband absorption in the TPU because the oscillator strength of the second photon absorption is proportional to the electron density in the initial state of the second transition. One appealing advantage for TPU-SCs compared with conventional IBSCs is the small spatial overlap of the quasi-Fermi level of electrons in the NGS and WGS. Generally, the quasi-Fermi level of the IB spatially overlaps with the quasi-Fermi level of the CB, causing the IB to thermally couple with the CB, resulting in a reduction in the output voltage. In contrast to such conventional IBCSs, the hetero-interface which boosts the voltage in a TPU-SC is separated from the portion generating additional current by absorbing below-gap photons in the WGS. This structure prevents energy relaxation from the CB of the WGS to that of the NGS.

Concept of TPU-SC and model used in calculation.
Our calculation is based on the detailed-balance framework which was originally proposed by Shockley and Queisser in ref. 1 . As described in ref. 1 , the detailed-balance framework considers a steady state between the carrier generation and recombination at the optimum operation point of a SC. This model has been widely used to calculate the ideal conversion efficiency of a SC. Here, we ignore nonradiative processes in the SC for predicting the ideal conversion efficiency limit. The total photon emission flux, N, with the energy range between E min and E max can be calculated using the generalised Planck equation incorporating the effect of the chemical potential, μ 4,24 : respectively. μ WGS and μ NGS are the quasi-Fermi level separations in the WGS and NGS, respectively, and μ up is the quasi-Fermi level separation due to TPU. ΔE c and ΔE v are the CB and VB discontinuity, respectively. G WGS and G NGS are the carrier-generation rates in the WGS and NGS, respectively, and G up is the carrier-generation rate due to up-conversion. R WGS , R NGS , and R up are the carrier-recombination rates in the WGS, NGS, and at the hetero-interface, respectively.
where T is the temperature, h is Planck's constant, c is the speed of light, and k b is Boltzmann's constant. By using equation (1), the generation rates of G WGS in the WGS, G NGS in the NGS, and G up for TPU can be expressed by: where X is the solar concentration factor, = . × − f 2 16 10 sun 5 is the solid angle of the sun, = T 6,000 K sun is the temperature of sun, E WGS and E NGS are the bandgap energies of the WGS and NGS, respectively, ΔE c is the CB discontinuity between the WGS and NGS, and = T 300 K cell is the temperature of the SC. Here, we considered that G up takes place solely within the NGS and then the carriers are transferred to the WGS. However, in the practical TPU-SC, the intraband excitation of the wavefunction component of accumulated electrons penetrating into the WGS barrier is strong. Conversely, the wavefunction component located in the NGS region should be solely excited within it, and the carriers are transferred to the WGS in the internal electric field. The relation between E WGS , E NGS , ΔE c , and VB discontinuity, ΔE v , can be given by: Each recombination rate can be given by: NGS N GS WGS c ell NGS and: where μ WGS and μ NGS are the quasi-Fermi level separations in the WGS and NGS, respectively, and μ up is the quasi-Fermi level separation due to TPU (see Fig. 1). We do not consider the surface recombination at the WGS/ NGS hetero-interface in this study. Here, we take into consideration electrons accumulated at the hetero-interface and the TPU occurring in the CB of the NGS. A similar process transpires when holes are accumulated at the hetero-interface and TPU occurs in the VB. In this case, ΔE c in equations (7) and (8) is replaced by ΔE v . According to these relations, the total current, J, generated in the TPU-SC can be obtained by: where q is the electronic charge. In the TPU-SC, the following current matching condition of TPU must be satisfied: The output voltage of the TPU-SC is given by

WGS N GS up
Finally, the total electrical power generated in TPU-SC is calculated as a product of VJ and, hence, the expected conversion efficiency can be estimated by the division of VJ by the total incident photon energy.

Conversion efficiency and recombination rate.
We calculated the conversion efficiency as a function of the NGS bandgap, E NGS , for different VB discontinuities, ΔE v . Figure 2a presents the results under one-sun illumination. Here, the bandgap of WGS, E WGS , is assumed to be 1.8 eV. As ΔE v is fixed in this calculation, ΔE c decreases with E NGS . When E NGS is 0 eV, the efficiency becomes equivalent to that of the SJSC with the same bandgap as ΔE c because ΔE v corresponds to a voltage loss. The efficiency increases with E NGS because of the voltage boost accomplished by TPU at the hetero-interface. Then, the conversion efficiency reaches a maximum of 43.6% at an E NGS of 0.63 eV when ΔE v = 0. This value coincides with the efficiency calculated for the well-known ideal IBSC with the same bandgap of 1.8 eV for the host semiconductor. The value of E NGS , at which the conversion efficiency exhibits the peak, shifts with varying ΔE v . An increasing ΔE v leads to a monotonic decrease in the peak efficiency because ΔE v leads to a voltage loss at the hetero-interface. This indicates that a zero-band discontinuity (ΔE v = 0) achieves a maximum conversion efficiency. Figure 2b shows detailed recombination rates, R NGS , for the interband transition in the NGS and energy relaxation rates, R up , for the intraband transition at the hetero-interface. When E NGS is small, R NGS is higher than R up by several orders of magnitude. As E NGS increases, R NGS decreases while R up increases and, finally, R up exceeds R NGS . When R up and R NGS coincide, the efficiency exhibits a peak. After reaching the maximum efficiency, R up becomes remarkably high, causing the efficiency to rapidly drop.

Effect of sunlight concentration and valence-band discontinuity on voltage boost. It is well
known that concentrator photovoltaics are one of the promising photovoltaic technologies which improve the conversion efficiency. Figure 3 displays the calculated results of the open-circuit voltage, V oc , as a function of sunlight concentration at various ΔE c -to-ΔE v ratios. Here, we choose E WGS and E NGS of 1.8 and 1.4 eV, respectively. In Fig. 3, the slopes of V oc for the TPU-SC are slightly steeper than that of the SJSC (blue line) at lower sunlight concentrations, which arises from the voltage-boost effect caused by TPU. At higher concentrations, all slopes coincide with that of the SJSC. The calculated quasi-Fermi splitting, μ up , at the hetero-interface at each ΔE c -to-ΔE v ratio is also shown in Fig. 3. With the increase in solar concentration, μ up increases and finally reaches ΔE c . As shown in Fig. 3, once μ up saturates, the slope of V oc becomes small. At lower concentrations, the increases in μ NGS and μ up contribute to the increase in V oc . Conversely, at higher concentrations, only the increase in μ NGS drives the increase in V oc , resulting in the shallow slope. The short-circuit current, on the other hand, completely coincides with that of the SJSC with a bandgap of 1.4 eV for any ΔE c -to-ΔE v ratio, as the current in the TPU-SC band lineup is limited by the carrier generation rate in the NGS.
Next, we focus on the effect of VB discontinuity on thes conversion efficiency. We systematically calculated the efficiency at different ΔE c -to-ΔE v ratios. Figure 4a,b present the results as a function of E WGS under one-sun and maximum concentration, respectively. In this calculation, E NGS is optimised to maximise the efficiency when calculating the value at each E WGS . When E WGS is small, ΔE c becomes small, and the calculated efficiency coincides with that of the SJSC indicated by the blue line, because the voltage boost effect becomes small in such a small band discontinuity. The efficiency of the TPU-SC increases with E WGS . Then, the calculated efficiency curve reaches a maximum. With the increase in ΔE v , the optimum E WGS exhibiting the maximum increases and its peak efficiency decreases. Here, it is noted that the efficiency curve given at ΔE v = 0 completely coincides with the result for an ideal IBSC proposed in ref. 6 . As the increase in ΔE v causes voltage loss at the hetero-interface, the efficiency decreases and, therefore, the optimum E WGS tends to become wide, improving the role of the voltage boost.   30 . It is noted that the band discontinuity depends on the material system. Even if the combination of the bandgaps comprising the hetero-interface is suitable for the TPU-SC, the conversion efficiency is also influenced by the band discontinuity. Figure 6 shows the maximum conversion efficiency as a function of sunlight concentration. The solid and dashed lines indicate the efficiencies  for the TPU-SCs with ΔE c -to-ΔE v ratios of 1:0 and 3:2, respectively. According to these calculations, approximately 10 (80) equivalent suns are necessary to surpass an efficiency of 50% with the ΔE c -to-ΔE v ratio of 1:0 (3:2).

Discussion
The TPU-SC is one practical realisation of the idea of photon ratchet [21][22][23] . Therefore, conceptually, the TPU-SC and the ratchet-band IBSC are the same. However, in the band diagram, there exists a difference between the TPU-SC and the ratchet-band IBSC. For the ratchet-band IBSC, the output voltage corresponds to the quasi-Fermi splitting between electrons in the CB and holes in the VB of the host material. On the other hand, the output voltage of TPU-SC corresponds to the difference of quasi-Fermi levels between electrons in the CB of WGS and holes in the VB of NGS. This difference appears as a difference in the calculation framework. For the  ratchet-band IBSC, the relationship among the bandgap of the host material, E g , the energy difference from the VB to IB, E VI , from the ratchet band to CB, E RC , and the IB to ratchet band, ΔE, is as follows: Equation (12) is corresponds to Eq. (5) for the TPU-SC, and E g , E VI , E RC , and ΔE correspond to E WGS , E NGS , ΔE c , and ΔE v , respectively. Here, ΔE in Eq. (12) must be negative because photo-generated electrons in the IB is necessary to relax towards the ratchet band. Conversely, for the TPU-SC, ΔE v in Eq. (5) is better to be positive because photo-generated holes in the WGS is easy to drift towards the NGS. The hetero-interface that boosts the voltage in a TPU-SC is separated from the portion generating additional current by absorbing below-gap photons in the WGS. This structure prevents energy relaxation from the CB of the WGS to that of the NGS, playing a similar role to that of the IB. Excited electrons and holes in the NGS are promptly separated in the internal electric field. Thereby, long-lived, high-density electrons accumulated in the depletion layer formed at the hetero-interface enables effective intraband excitation and accomplishes efficient TPU. The effect of the voltage boost decreases with ΔE v . Therefore, the conversion efficiency of the TPU-SC does not exceed the ideal value of the IBSC. However, the TPU-SC does not require any complicated quantum structures and is a SJSC containing a simple hetero-interface. Thus, we can expect bulk quality carrier transport in TPU-SCs. TPU-SCs have similar structure to dual-junction SCs. Dual-junction solar cells comprise a series of p-n junctions connected through a tunnelling junction, whereas a TPU-SC is a SJSC. Thus, TPU does not occur in a dual-junction SC, and its conversion efficiency is smaller than that of a TPU-SC, as expected in Fig. 6. Moreover, current matching between the WGS and NGS is not required in TPU-SCs. This advantage causes TPU-SCs to be robust against changes in the sunlight spectrum 31,32 . These results observed in the TPU-SC offer a route to high-conversion-efficiency solar cells exceeding 50%.

Methods
We performed calculations using Visual Studio Community 2015 and GNU Scientific Library. The programming language used was Visual C++.
Code availability. The computer code used in this study are available from the corresponding author upon request.
Data Availability. The data that support the findings of this study are available from the corresponding author upon request.