**2. Analysis of non-radiative recombination and resistance losses of single-junction solar cells**

By using our analytical model [8, 9], potential efficiencies of various solar cells are discussed. This model considers the efficiency loss such as non-radiative recombination and resistance losses, which are reasonable assumption because conventional solar cells often have a minimal optical loss. The non-radiative recombination loss is characterized by external radiative efficiency (ERE), which is the ratio of radiatively recombined carriers against all recombined carriers. In other words, we have ERE = 1 at Shockley-Queisser limit [5]. EREs of state-of-the-art solar cells can be found in some publications such as references [2, 10–13]. In this chapter, the EREs of various solar cells are estimated by the following relation [14]:

$$\mathbf{V\_{oc}} = \mathbf{V\_{oc:rad}} + (\mathbf{kT/q})\ln(\mathbf{ERE}),\tag{2}$$

where Voc the measured open-circuit voltage, k the Boltzmann constant, T the temperature, and q the elementary charge. Voc:rad the radiative open-circuit voltage and is expressed by the following Eq. [15]

$$\mathbf{V}\_{\rm oc,rad} = (\mathbf{kT/q}) \ln \left[ \mathbf{J}\_{\rm ph} \right] \mathbf{V}\_{\rm oc,rad} / \left[ \mathbf{J}\_{\rm o,rad} + \mathbf{1} \right],\tag{3}$$

where [Jph]Voc,rad is the photocurrent at open-circuit in the case when there is only radiative recombination and Jo,rad the saturation current density in the case of radiative recombination.

0.28 V for Eg/q - Voc;rad value reported in [15–17] were used in our analysis. Where Eg is the bandgap energy. The second term on the right-hand side of Eq. (2) is denoted as Voc;nrad, the voltage-loss due to non-radiative recombination and is expressed by the following Eq. [15].

$$
\Delta \mathbf{V}\_{\rm oc;nrad} = \mathbf{V}\_{\rm oc;rad} - \mathbf{V}\_{\rm oc} = (\mathbf{k} \mathbf{T}/\mathbf{q}) \ln \left[ \mathbf{J}\_{\rm rad}(\mathbf{V}\_{0c})/\mathbf{J}\_{\rm rec}(\mathbf{V}\_{\rm oc}) \right] = -(\mathbf{k} \mathbf{T}/\mathbf{q}) \ln \left( \mathbf{ERE} \right), \tag{4}
$$

where Jrad(V0c) is the radiative recombination current density and Jrec(Voc) is the non-radiative recombination current density.

**Figure 3** shows open-circuit voltage drop compared to band gap energy (Eg/q – Voc) and non-radiative voltage loss (Voc,nrad) in GaAs, InP, AlGaAs and

#### **Figure 3.**

*Open-circuit voltage drop compared to band gap energy (Eg/q – Voc) and non-radiative voltage loss (Voc,nrad) in GaAs, InP, AlGaAs and InGaP solar cells as a function of ERE.*

InGaP solar cells [2, 8–13, 17] as a function of ERE. High ERE values of 22.5% and 8.7% have been observed for GaAs and InGaP, respectively compared to InP (0.1%) and AlGaAs (0.01%).

The resistance loss of a solar cell is estimated solely from the measured fill factor. The ideal fill factor FF0, defined as the fill factor without any resistance loss, is estimated by [18].

$$\text{FF}\_0 = (\mathbf{v}\_{\text{oc}} - \ln\left(\mathbf{v}\_{\text{oc}} + \mathbf{0} : \mathbf{71}\right)) / (\mathbf{v}\_{\text{oc}} + \mathbf{1}),\tag{5}$$

where voc is

$$\mathbf{v}\_{\rm oc} = \mathbf{V}\_{\rm oc} / (\mathbf{nkT/q}).\tag{6}$$

**Figure 4.** *Correlation between fill factor and resistance loss in GaAs, InP, AlGaAs and InGaP solar cells.*

The measured fill factors can then be related to the series resistance and shunt resistance by the following Eq. [18]:

$$\text{FF} \approx \text{FF}\_0 (\mathbf{1} - \mathbf{r}\_s) \left(\mathbf{1} - \mathbf{r}\_{\text{sh}}\right)^{-1} \approx \text{FF}\_0 \left(\mathbf{1} - \mathbf{r}\_{\text{s}} \mathbf{-r}\_{\text{sh}}\right)^{-1}) = \text{FF}\_0 (\mathbf{1} - \mathbf{r}),\tag{7}$$

where rs is the series resistance, and rsh is the shunt resistance normalized to RCH. The characteristic resistance RCH is defined by [18]

$$\mathbf{R\_{CH}} = \mathbf{V\_{oc}} / \mathbf{J\_{sc}},\tag{8}$$

r is the total normalized resistance defined by r = rs + rsh�<sup>1</sup> .

**Figure 4** shows correlation between fill factor and resistance loss [2, 8–13, 17] in GaAs, InP, AlGaAs and InGaP solar cells. Lower resistance losses of 0.01-0.03 have been realized for GaAs, InP and InGaP solar cells compared to 0.05 for AlGaAs.
