**2. Generalized detailed balance model applied to optically-thin nanoenhanced devices**

The concept of detailed balance is often used to estimate the limiting efficiency of ideal photovoltaic devices. Since its introduction by Shockley-Queisser, detailed balance calcula‐ tions have been generalized to include a continuous absorbance function and a variety of different cell geometries [9-11]. More recently, detailed balance concepts have been further generalized and applied to the analysis of experimental results from several different types of functional photovoltaic devices [12-14]. In this section, a generalized detailed balance model is applied to optically-thin nano-enhanced photovoltaic absorber structures. A device geom‐ etry with negligible photon recycling is considered the starting point for the discussion because of the reduced absorption inherent to optically-thin structures. Experimental dark current and external quantum efficiency characteristics from a high-voltage InGaAs quantum well structure are then analyzed within the framework of this generalized detailed balance model.

By extending detailed balance concepts to include both carrier generation and carrier transport properties, Kirchartz and Rau have demonstrated that the photovoltaic external quantum efficiency, Qe (E), measured at normal incidence, can be related to the radiative dark current and luminescent characteristics of photovoltaic (PV) and light emitting diode (LED) devices [13]. The radiative dark current (Jrad) follows an n=1 voltage dependence, β (V) = exp(qV/kT) – 1, that will fundamentally limit the operating voltage of a photovoltaic device, and that can be related to the experimentally measured external quantum efficiency:

$$\mathbf{J}\_{\rm rad} = \mathbf{q} \{ \mathbf{Q}\_{\epsilon}(E) \: \phi\_{bb}(E) \: \boldsymbol{\beta}(V) \: \boldsymbol{E} \} \, \text{d}E \tag{1}$$

where q is the elementary charge, V is the applied bias voltage, and kT is the thermal energy. The relevant blackbody spectral photon density spectrum, ϕbb (E), in Equation (1) is contingent upon the cell geometry, but can be generalized here with the introduction of a dark current factor (Fdc) to characterize, among other things, the overall effective etendue of the photovoltaic device, such that:

$$\phi\_{bb}(E) = \left\langle 2 \, F\_{dc} \, \Big|\, h^3 \, c^2 \right\rangle E^2 \left/ \left\langle \exp\left\langle \text{E}/kT \right\rangle - 1\right\rangle \right. \tag{2}$$

In many detailed balance calculations, Fdc is often assumed to be as low as π, corresponding to the limit in which emitted photons are reabsorbed after perfect reflection off the back surface or total internal reflection off the front surface. However, such photon recycling effects, which effectively restrict the angular emissions of the device, may become negligible in optically-thin structures with low reflectance surfaces and/or high absorbing contact layers. In this limit of negligible re-absorption:

$$F\_{dc} = \text{2 }\text{\textbullet }\text{\textbullet }\text{\textbullet }^2\text{\textbullet}\tag{3}$$

where nb is the refractive index of the barrier material surrounding the optically-thin absorber.

The external quantum efficiency of a photovoltaic device, Qe (E), is also related to the short circuit current density (Jsc) via:

$$\mathbf{J}\_{sc} = \mathbf{q}[\mathbf{L}\_c(\mathbf{E}) \; \mathbf{Q}\_e(\mathbf{E}) \; \boldsymbol{\phi}\_{sun}(\mathbf{E}) \; dE \tag{4}$$

where ϕsun (E) is the incident solar spectrum and Uc (E) is a correction factor to the photocurrent introduced here in order to account for optical up-conversion effects [15]. In conventional PV devices, Uc (E) = 1. However, in some devices, including optically-thin nano-enhanced absorber structures, the presence of low-energy photons can enhance carrier generation beyond that measured in a typical photovoltaic external quantum efficiency measurement [16-18], resulting in an effective enhancement in the incident solar spec‐ trum such that Uc (E) > 1.

functional photovoltaic devices [12-14]. In this section, a generalized detailed balance model is applied to optically-thin nano-enhanced photovoltaic absorber structures. A device geom‐ etry with negligible photon recycling is considered the starting point for the discussion because of the reduced absorption inherent to optically-thin structures. Experimental dark current and external quantum efficiency characteristics from a high-voltage InGaAs quantum well structure are then analyzed within the framework of this generalized detailed balance model.

By extending detailed balance concepts to include both carrier generation and carrier transport properties, Kirchartz and Rau have demonstrated that the photovoltaic external quantum efficiency, Qe (E), measured at normal incidence, can be related to the radiative dark current and luminescent characteristics of photovoltaic (PV) and light emitting diode (LED) devices [13]. The radiative dark current (Jrad) follows an n=1 voltage dependence, β (V) = exp(qV/kT) – 1, that will fundamentally limit the operating voltage of a photovoltaic device, and that can be

related to the experimentally measured external quantum efficiency:

Jrad =q*∫Qe*

(*E*) <sup>=</sup> (2 *Fdc* / *<sup>h</sup>* <sup>3</sup> *<sup>c</sup>* <sup>2</sup>

Jsc=q*∫Uc*

device, such that:

250 Solar Cells - New Approaches and Reviews

negligible re-absorption:

circuit current density (Jsc) via:

ϕ*bb*

(*E*) ϕ*bb*

where q is the elementary charge, V is the applied bias voltage, and kT is the thermal energy. The relevant blackbody spectral photon density spectrum, ϕbb (E), in Equation (1) is contingent upon the cell geometry, but can be generalized here with the introduction of a dark current factor (Fdc) to characterize, among other things, the overall effective etendue of the photovoltaic

In many detailed balance calculations, Fdc is often assumed to be as low as π, corresponding to the limit in which emitted photons are reabsorbed after perfect reflection off the back surface or total internal reflection off the front surface. However, such photon recycling effects, which effectively restrict the angular emissions of the device, may become negligible in optically-thin structures with low reflectance surfaces and/or high absorbing contact layers. In this limit of

*Fdc* =2 *π nb*

(*E*) *Qe*

where nb is the refractive index of the barrier material surrounding the optically-thin absorber.

The external quantum efficiency of a photovoltaic device, Qe (E), is also related to the short

where ϕsun (E) is the incident solar spectrum and Uc (E) is a correction factor to the photocurrent introduced here in order to account for optical up-conversion effects [15]. In

(*E*) *β*(*V* ) *dE* (1)

) *E* <sup>2</sup> / (exp (E / kT) -1) (2)

<sup>2</sup> (3)

(*E*) ϕ*sun*(*E*) *dE* (4)

The validity of the generalized detailed balance model for optically-thin photovoltaic devices described above has been confirmed by analyzing experimental results from a highvoltage InGaAs quantum well solar cell. The measured external quantum efficiency and current-voltage characteristics from a high-voltage GaAs-based diode with a single nearlysquare 15 nm InGaAs well embedded within the junction depletion region are summar‐ ized in Figure 2. The baseline diode consists of an extended p-type wide band gap emitter, which minimizes non-radiative recombination, and a relatively thin (0.5 μm) GaAs base layer, similar to the high-voltage InGaAs well structures described in reference [19]. Asgrown wafers were quartered and 0.25 cm2 thin-film cells were fabricated using a sub‐ strate removal process at MicroLink Devices [20]. The measured external quantum efficiency characteristics summarized in Figure 2 (a) were taken from a cell employing a low reflectance two-layer anti-reflectance coating on the front surface and an absorbing Gebased back metallization. Small, simple mesa test devices were also fabricated via stand‐ ard wet etch chemistry and photolithography on a second quarter piece of the same wafer. For the dark I-V measurements reported in Figure 2 (b), a test structure consisting of a device with a junction area of 200 μm x 270 μm has been employed.

The measured dark current in Figure 2 (b) exhibits a non-ideal voltage dependence at lower bias, presumably due to non-radiative recombination within the depletion region. However, the voltage dependence of the dark current approaches unity as the bias approaches 1 V. Quantitatively, the measured dark current at higher bias nearly matches the radiative dark current calculated from Equations (1) – (3), assuming T = 25o C, nb = 3.5 (refractive index of the GaAs surrounding the well), and the measured external quantum efficiency shown in Figure 2 (a). While the calculated n=1 component of the dark current shown in Figure 2 (b) assumes negligible re-absorption (Fdc = 2πn<sup>b</sup> 2 ), we note that the dark current factor can be generalized further to account for non-radiative and non-equilibrium effects. Non-radiative recombination will result in higher effective Fdc values that can also vary with voltage if recombination occurs within the junction depletion region. On the other hand, lower Fdc values could be obtained if the overall angular emissions of the device are restricted, for example via photon recycling, or through non-equilibrium effects such as hot carrier extraction. Finally, we note that mislead‐ ingly high Fdc values could be obtained if the falloff in Qe (E) is not accurately measured at longer wavelengths, as emissions from the Urbach tail region can have a significant impact on the dark current, as will be discussed in later sections.

were quartered and 0.25 cm2

the incident solar spectrum such that Uc (E) > 1.

presence of low-energy photons can enhance carrier generation beyond that measured in a typical photovoltaic external quantum efficiency measurement [16-18], resulting in an effective enhancement in

The validity of the generalized detailed balance model for optically-thin photovoltaic devices described above has been confirmed by analyzing experimental results from a high-voltage InGaAs quantum well solar cell. The measured external quantum efficiency and current-voltage characteristics from a highvoltage GaAs-based diode with a single nearly-square 15 nm InGaAs well embedded within the junction depletion region are summarized in Figure 2. The baseline diode consists of an extended p-type wide band gap emitter, which minimizes non-radiative recombination, and a relatively thin (0.5 m) GaAs base layer, similar to the high-voltage InGaAs well structures described in reference [19]. As-grown wafers

MicroLink Devices [20]. The measured external quantum efficiency characteristics summarized in Figure 2 (a) were taken from a cell employing a low reflectance two-layer anti-reflectance coating on the front surface and an absorbing Ge-based back metallization. Small, simple mesa test devices were also fabricated via standard wet etch chemistry and photolithography on a second quarter piece of the same

thin-film cells were fabricated using a substrate removal process at

InGaAs quantum well solar cell structure employing an extended wide band-gap AlGaAs/InGaP emitter heterojunction. The dashed line in (b) is the calculated Jrad from equations (1) – (3) using the measured Qe (E). **Figure 2.** External quantum efficiency spectra (a) and dark current-voltage measurements (b) from a high voltage In‐ GaAs quantum well solar cell structure employing an extended wide band-gap AlGaAs/InGaP emitter heterojunction. The dashed line in (b) is the calculated Jrad from equations (1) – (3) using the measured Qe (E).

**Figure 2:** External quantum efficiency spectra (a) and dark current-voltage measurements (b) from a high voltage

### **3. Dependence of radiative dark current on absorber thickness** 5

Like conventional photovoltaic devices, the output voltage of optically-thin solar cells is governed by the underlying dark current. The dark current of typical III-V semiconductor diodes is composed of several different components involving both radiative and nonradiative processes. Non-radiative dark current processes can follow either an n=2 or n=1 voltage dependence, contingent upon the location of the recombination. Non-radiative defects within the junction depletion region contribute to the n=2 space charge recombination component of the dark current, while non-radiative defects in the quasi-neutral regions of the device contribute to the n=1 component. Radiative processes also follow an n=1 voltage dependence, and as discussed in the previous section, will ultimately limit the voltage performance of the best photovoltaic devices. In this section, the impact of absorber layer thickness on the radiative dark current is detailed after making some simplifying assumptions regarding absorption processes in GaAs-based devices.

In a previous section, we summarized how detailed balance considerations can be used to relate the radiative dark current to the external quantum efficiency – e.g. Equation (1). The external quantum efficiency is in turn proportional to the effective absorber layer width (Weff) and the absorption coefficient α (E):

$$\mathcal{Q}\_{\mathbf{e}}(E) = \mathcal{C}(E) \, a(E) \, \mathcal{W}\_{\mathbf{e}\overline{\mathbf{f}}} \tag{5}$$

where C (E) is a correction factor which accounts for loss mechanisms, such as reflections off the front surface and photogenerated carrier recombination prior to collection. While the absorption coefficient of GaAs is arguably the most well-known of all the III-V compounds, the reported values can vary significantly with doping, particularly at energies close to and below the band gap. In this section, the GaAs absorption coefficient is modeled using a piecewise continuous function as described in Miller *et al.* [21] but calibrated using the experimental data from Kurtz *et al.* [22], as shown in Figure 3 (a). thickness (Wp) via a three-dimensional radiative recombination coefficient (B3D): (6) J������ � ���� ������ � <sup>⁄</sup> ���� where r is Asbeck's photon recycling co-factor [23-24]. Figure 3 (d) summarizes the dependence of <sup>r</sup> on thickness, as inferred from fitting Equation (6) to the radiative dark current derived from detailed balance calculations as described above and summarized in Figure 3 (c). Assuming ni = 2.1 x 106 cm-3, a typical value quoted for GaAs, the fit also implies B3D = 3.9 x 10-10 cm3 /s, reasonably in-line with previous estimates of the radiative recombination coefficient in GaAs [23].

spectrum of a 50 nm GaAs layer. The Qe (E) generated from Equation (5) can then be combined with

current density (Jo1,rad) can be related to the intrinsic carrier density (ni) and the physical absorber layer

**3. Dependence of radiative dark current on absorber thickness**

The dashed line in (b) is the calculated Jrad from equations (1) – (3) using the measured Qe (E).

regarding absorption processes in GaAs-based devices.

and the absorption coefficient α (E):

Like conventional photovoltaic devices, the output voltage of optically-thin solar cells is governed by the underlying dark current. The dark current of typical III-V semiconductor diodes is composed of several different components involving both radiative and nonradiative processes. Non-radiative dark current processes can follow either an n=2 or n=1 voltage dependence, contingent upon the location of the recombination. Non-radiative defects within the junction depletion region contribute to the n=2 space charge recombination component of the dark current, while non-radiative defects in the quasi-neutral regions of the device contribute to the n=1 component. Radiative processes also follow an n=1 voltage dependence, and as discussed in the previous section, will ultimately limit the voltage performance of the best photovoltaic devices. In this section, the impact of absorber layer thickness on the radiative dark current is detailed after making some simplifying assumptions

5

**Figure 2.** External quantum efficiency spectra (a) and dark current-voltage measurements (b) from a high voltage In‐ GaAs quantum well solar cell structure employing an extended wide band-gap AlGaAs/InGaP emitter heterojunction.

 **(a) (b) Figure 2:** External quantum efficiency spectra (a) and dark current-voltage measurements (b) from a high voltage InGaAs quantum well solar cell structure employing an extended wide band-gap AlGaAs/InGaP emitter heterojunction. The dashed line in (b) is the calculated Jrad from equations (1) – (3) using the measured Qe (E).

**Current**

**Density**

**(mA/cm2)**

**1.0E‐04**

**0.60 0.70 0.80 0.90 1.00 1.10**

**Voltage (V)**

**1.0E‐03**

**1.0E‐02**

**1.0E‐01**

**1.0E+00**

**1.0E+01**

**1.0E+02**

presence of low-energy photons can enhance carrier generation beyond that measured in a typical photovoltaic external quantum efficiency measurement [16-18], resulting in an effective enhancement in

The validity of the generalized detailed balance model for optically-thin photovoltaic devices described above has been confirmed by analyzing experimental results from a high-voltage InGaAs quantum well solar cell. The measured external quantum efficiency and current-voltage characteristics from a highvoltage GaAs-based diode with a single nearly-square 15 nm InGaAs well embedded within the junction depletion region are summarized in Figure 2. The baseline diode consists of an extended p-type wide band gap emitter, which minimizes non-radiative recombination, and a relatively thin (0.5 m) GaAs base layer, similar to the high-voltage InGaAs well structures described in reference [19]. As-grown wafers

MicroLink Devices [20]. The measured external quantum efficiency characteristics summarized in Figure 2 (a) were taken from a cell employing a low reflectance two-layer anti-reflectance coating on the front surface and an absorbing Ge-based back metallization. Small, simple mesa test devices were also fabricated via standard wet etch chemistry and photolithography on a second quarter piece of the same wafer. For the dark I-V measurements reported in Figure 2 (b), a test structure consisting of a device with

thin-film cells were fabricated using a substrate removal process at

the incident solar spectrum such that Uc (E) > 1.

a junction area of 200 m x 270 m has been employed.

**300 500 700 900**

**Wavelength (nm)**

were quartered and 0.25 cm2

**0.01**

**0.1**

**Quantum**

**Efficiency (%)**

**1**

**10**

**100**

252 Solar Cells - New Approaches and Reviews

In a previous section, we summarized how detailed balance considerations can be used to relate the radiative dark current to the external quantum efficiency – e.g. Equation (1). The external quantum efficiency is in turn proportional to the effective absorber layer width (Weff)

**Figure 3:** (a) Modeled (line) and measured (circles) absorption spectra of GaAs; (b) modeled external quantum efficiency spectra for a 50 nm GaAs absorber; (c) dependence of the radiative saturation dark current as a function **Figure 3.** (a) Modeled (line) and measured (circles) absorption spectra of GaAs; (b) modeled external quantum efficien‐ cy spectra for a 50 nm GaAs absorber; (c) dependence of the radiative saturation dark current as a function of absorber layer thickness as calculated (circles) from detailed balance considerations and fit (line) to Equation (6); and (d) photon recycling factor as a function of absorber thickness, derived from the fit to Equation (6).

7 In the limit of negligible photon recycling, the effective absorber layer width can be equated to the physical absorber layer width (Weff = Wp). Furthermore, in ideal structures, reflection losses are minimized and all of the absorbed photons are collected, e.g. C (E) = 1. With these simplifying assumptions, the external quantum efficiency can be calculated from equation (5) for any given thickness of an ideal GaAs absorber. For example, Figure 3 (b) depicts the calculated external quantum efficiency spectrum of a 50 nm GaAs layer. The Qe (E) generated from Equation (5) can then be combined with equations (1) – (3) to calculate the dependence of saturated radiative dark current on absorber thickness for GaAs, as summarized in Figure 3 (c). Clearly the radiative dark current can be reduced, and the operating voltage enhanced, by minimizing the absorber layer thickness.

Radiative recombination can also be described mechanistically in terms of carrier recombina‐ tion via the use of a radiative recombination coefficient. With this mechanist approach, the radiative saturation current density (Jo1,rad) can be related to the intrinsic carrier density (ni ) and the physical absorber layer thickness (Wp) via a three-dimensional radiative recombination coefficient (B3D):

$$\mathbf{J}\_{\rm o1,rad} = \mathbf{q} \ n\_{\rm i}^2 \left( \mathbf{B}\_{\rm 3D} / \phi\_{\rm r} \right) \mathbf{W}\_{\rm p} \tag{6}$$

where ϕ<sup>r</sup> is Asbeck's photon recycling co-factor [23-24]. Figure 3 (d) summarizes the depend‐ ence of ϕr on thickness, as inferred from fitting Equation (6) to the radiative dark current derived from detailed balance calculations as described above and summarized in Figure 3 (c). Assuming ni = 2.1 x 106 cm-3, a typical value quoted for GaAs, the fit also implies B3D = 3.9 x 10-10 cm3 /s, reasonably in-line with previous estimates of the radiative recombination coefficient in GaAs [23].
