**5. Impact of enhanced optical path length**

densities within the quantum well. In the limit of evenly emitting wells in which the effective carrier densities are the same within each well, the radiative current density generated by a

where nqw and pqw are the effective quantum well electron and hole densities per unit area at zero bias and Mqw is the number of wells in the structure (Mqw=1in the set considered here). For any given effective well energy, equation (7) implies that the radiative component of the

 **(a) (b) Figure 4:** Normalized photoluminescence spectra (a) and dark current-voltage measurements (b) from a set of highvoltage InGaAs quantum well solar cell structures, all emitting at approximately 1.325 eV but with varying well

**Figure 4.** Normalized photoluminescence spectra (a) and dark current-voltage measurements (b) from a set of high-volt‐ age InGaAs quantum well solar cell structures, all emitting at approximately 1.325 eV but with varying well thickness.

In Figure 5, the reverse saturation current density values inferred from both the dark diode current and photoluminescence measurements are compared to calculations using Equations (6) and (7). For the thicker samples, the variation in Jo1,rad with absorber layer thickness is well fit by a 3D representation of the carrier recombination, as given by Equation (6), assuming the radiative recombination coefficient is independent of indium composition (B3D = 3.9 x 10-10

/s) and that the intrinsic carrier combination scales inversely with the effective energy gap

variation in Jo1,rad with absorber layer thickness is well fit by a 2D representation of the carrier recombination as given by Equation (4), assuming nqw = pqw = 0.37 cm-2 and B2D = 1.3 x 10-3 cm2

1/2 exp(-Eg/2kT) = 1.57 x 107 cm-3). On the other hand, for the thinner samples, the

**1.0E-04**

**0.65 0.75 0.85 0.95 1.05**

**Voltage (V)**

**n = 1**

/s.

**1.0E-03**

**1.0E-02**

**1.0E-01**

**Current Density (mA/cm2)**

**1.0E+00**

**1.0E+01**

**1.0E+02**

11

**Figure 5:** Reverse saturation current density of the n=1 component of the diode current as derived from the measured dark diode current (solid circles) and calculated from the measured PL spectra (open diamonds). Also shown is the expected variation in the radiative dark current in the 3D and 2D regimes using Equations (6) and (7).

dark current will scale with the number of wells, independent of well thickness.

**30 nm 22.5 nm 15 nm 2.5 nm**

Jo1,rad =q nqw pqw B2DMqw (7)

**30 nm 22.5 nm 15 nm 2.5 nm**

**n = 2**

multiple quantum well structure can be expressed as:

thickness.

cm3

(ni

= (NcNv)

**0.0**

**1.20 1.30 1.40 1.50**

**PL Energy (eV)**

**0.5**

**1.0**

**1.5**

**PL Signal (a.u.)**

**2.0**

**2.5**

256 Solar Cells - New Approaches and Reviews

The application of light trapping structures to thin-film devices provides a means to both further suppress the radiative dark current via photon recycling and increase the current output via enhanced optical path lengths within the thin absorber structure. Figure 6 summa‐ rizes the current-voltage characteristics of a simple example of an optically-thin absorber structure employing a reflective back contact [6]. This uncoated test device employs a 30 nm GaAs absorber embedded within a wider energy-gap InGaP/AlGaAs heterojunction, and has been fabricated into a thin-film device using an epitaxial liftoff process at MicroLink Devices [20]. Record-low dark current characteristics for a GaAs-based device have resulted in an ultrahigh open circuit voltage (Voc) of 1.122 V at a short circuit current density (Jsc) of 14.7 mA/cm2 .

Equation (6) can be employed to estimate the expected impact of enhanced optical path length on the radiative dark current of a GaAs-based device. In particular, Figure 7 compares the calculated dependence of the radiative n=1 saturation current density as a function of absorber thickness for four different structures with varying optical path length (OPL) enhancements. Because photon emissions are omnidirectional in nature, the OPL enhancements assumed in Figure 7 represent angle averaged values. For these calculations, we further assume that B = 3.9 x 10-10 cm3 /s, ni = 2.1 x 106 cm-3, and that self- absorption effects – see ϕr dependence on thickness shown in Figure 3 (d) – scale with the OPL factor. Enhancements in the OPL due to reflections off the front and back surfaces of the device can result in the re-absorption of emitted photons and thus a significant increase of self-absorption effects, particularly in thicker absorber structures.

**Figure 6.** Illuminated current-voltage characteristics from a small area (0.25 cm2 ), optically-thin GaAs test device fabri‐ cated via epitaxial liftoff [19]. An optical photograph of the flexible test cells is shown inset.

For thin absorbers, such as the test device summarized in Figure 6, the impact of light trapping on radiative dark current is relatively small. However, the impact of light trapping on the short circuit current of thin-absorber structures can be quite significant. Figure 8 highlights the dependence of the short circuit current density on both the physical absorber layer thickness and the effective optical thickness due to enhancements in the optical path length. The OPL enhancements shown in Figure 8, unlike Figure 7, are not angle averaged but instead describe the average path length of normal incident photons. For these calculations, the GaAs absorp‐ tion coefficient was modeled using a piecewise continuous function described in Miller *et al.* [21] but calibrated using the experimental data from Kurtz *et al.* [22], as shown earlier in Figure 3 (a). After accounting for reflection off the front surface of an uncoated device (R ~ 35%), it was further assumed that all absorbed incident low energy photons with wavelengths greater than 715 nm generated collectable electron-hole pairs. Higher energy photons are assumed to be absorbed in a wider energy gap matrix surrounding the GaAs layer, providing 13.1 mA/cm2 of current under simulated AM1.5 illumination. The effective thickness in Equation (5) was then assumed to be the product of the physical thickness and the OPL factor. As seen in Figure 7, the application of light trapping structures which can enhance the optical path length of incident photons is projected to have a significant impact on the current output of thin absorber structures, but has minimal impact on thicker absorber structures.

photons and thus a significant increase of self-absorption effects, particularly in thicker

), optically-thin GaAs test device fabri‐

**Figure 6.** Illuminated current-voltage characteristics from a small area (0.25 cm2

cated via epitaxial liftoff [19]. An optical photograph of the flexible test cells is shown inset.

thin absorber structures, but has minimal impact on thicker absorber structures.

For thin absorbers, such as the test device summarized in Figure 6, the impact of light trapping on radiative dark current is relatively small. However, the impact of light trapping on the short circuit current of thin-absorber structures can be quite significant. Figure 8 highlights the dependence of the short circuit current density on both the physical absorber layer thickness and the effective optical thickness due to enhancements in the optical path length. The OPL enhancements shown in Figure 8, unlike Figure 7, are not angle averaged but instead describe the average path length of normal incident photons. For these calculations, the GaAs absorp‐ tion coefficient was modeled using a piecewise continuous function described in Miller *et al.* [21] but calibrated using the experimental data from Kurtz *et al.* [22], as shown earlier in Figure 3 (a). After accounting for reflection off the front surface of an uncoated device (R ~ 35%), it was further assumed that all absorbed incident low energy photons with wavelengths greater than 715 nm generated collectable electron-hole pairs. Higher energy photons are assumed to be absorbed in a wider energy gap matrix surrounding the GaAs layer, providing 13.1 mA/cm2 of current under simulated AM1.5 illumination. The effective thickness in Equation (5) was then assumed to be the product of the physical thickness and the OPL factor. As seen in Figure 7, the application of light trapping structures which can enhance the optical path length of incident photons is projected to have a significant impact on the current output of

absorber structures.

258 Solar Cells - New Approaches and Reviews

**Figure 7.** Projected dependence of the radiative saturation dark current as a function of GaAs absorber layer thickness assuming four different structures with varying degrees of light trapping, resulting in optical path length enhance‐ ments of 1x, 4x, 16x, and 24x. Also shown is the n=1 saturation dark current extracted from measurements on the opti‐ cally-thin GaAs test device summarized in Figure 6 and described in more detail in reference [6].

**Figure 8.** Projected dependence of the uncoated short circuit current as a function of GaAs absorber layer thickness under AM 1.5 illumination, assuming five different structures with varying degrees of light trapping, resulting in opti‐ cal path length enhancements of 1x, 2x, 4x, 8x, and 16x. Also shown is the short circuit current density measured on the optically-thin GaAs test structure characterized in Figure 6 and described in more detail in reference [6].
