1 Introduction

In solar cells, photon recycling (PR) describes the secondary generation of charge carriers due to re-absorption of photons that are emitted internally in a radiative recombination event. In order for PR to be of significance, the solar cell needs to operate close to the radiative limit and a steep absorption edge as well as a large absorption coefficient and a small Stokes shift between absorption and emission are required such as to allow for sufficient re-absorption. While these conditions are hardly met in most real solar cells, PR was shown to be instrumental for the achievement of record efficiencies in thin-film GaAs devices [1, 2]. Renewed interest in PR was spurred with the advent of the metal-halide perovskite absorber materials that largely fulfill the above requirements and indeed showed signs of re-absorption and PR [3,4,5,6,7].

The potential of PR for photovoltaic energy conversion has been assessed by optical simulation mostly. Early work on PR in III–V semiconductor solar cells [8, 9] was based on ray optics, which is appropriate in bulk devices. For thin films with thickness on the order of the wavelength of the light to be absorbed, on the other hand, wave optical effects such as interference and cavity modes call for models based on rigorous solution of Maxwell’s equations [10,11,12]. While ray-optical models for secondary generation have been integrated with electrical device simulation for propagation of the re-absorption into an enhancement of the open circuit voltage (\(V_{\text {OC}}\)) in a number of approaches [13,14,15], rigorous wave optical treatment of the re-absorption process in a full opto-electronic simulation of thin-film devices remains the exception [16, 17]. Furthermore, quantification of PR effects is often based on detailed-balance arguments [4, 10, 18, 19], comparing internal and external emission based on the generalized Planck [20] and Kirchhoff laws [21, 22]. However, the internal emission considered in the Van Roosbroeck–Shockley (VRS) detailed-balance formalism assumes an optically homogeneous medium, which is not consistent with the treatment of optical modes in either, the absorptance and the external emission [23].

The strong enhancement of the photon density of states (DOS) due to leaky-mode resonances and guided modes offers the prospect of a sizable \(V_{\text {OC}}\) increment at ultra-scaled absorber thicknesses below 100 nm [4]. This adds to the increased interest in ultra-thin absorber solar cells that target high efficiencies at vastly reduced material usage allowing for the utilization of less abundant materials and higher cost fabrication techniques as well as application in areas where light weight and flexibility are key [24,25,26]. In fact, efficiencies close to 20% and exceeding 22% were recently achieved with just 200 nm and 260 nm of GaAs on structured back reflectors [27, 28], respectively, and theoretical estimates of nanophotonic light trapping predict near complete light absorption in GaAs films with thickness below 25 nm [29,30,31].

The impact of cavity effects and surface plasmon polaritons on the optical modes of thin-film solar cells has been considered in the assessment of absorptance [32, 33]. Similarly, plasmonic losses at back reflectors and their mitigation via dielectric buffer layers have been studied primarily with regard to the scattering of the incident illumination [34], even though the quenching of luminescence at metallic electrodes is a well known issue in organic light emitting devices [35]. On the other hand, at ultra-scaled dimensions, the absorber starts to show finite size effects also in its electronic properties, as the contact regions extend over a sizable fraction of the device, and even small changes in the device configuration can strongly modify the opto-electronic properties [36]. An example of such deviations from electronically bulk-like behavior—which in fact invalidate the assumptions underlying most macroscopic drift-diffusion-type charge transport simulations—is the effects of strong built-in fields on the electronic density of states, absorptance and emission, and the corresponding dependence on the point of device operation, that were predicted theoretically using the approach underlying this work [36,37,38] and which have recently been observed also experimentally [39].

Here, we now provide a microscopic analysis of PR in ultra-thin absorber solar cells that not only includes the rigorous treatment of spontaneous emission into guided and leaky cavity modes and coupling to surface plasmon polaritons [40, 41] and considers the impact of mirror design on PR beyond the ray-optics limit [42,43,44], but also reflects the effects of finite size and built-in fields on the electronic structure and associated local opto-electronic material properties. As the comprehensive non-equilibrium quantum statistical mechanics framework used here to describe the dynamics of coupled electrons and photons treats quantum optical effects on equal footing with quantum transport, the impact of PR on photovoltaic performance in dependence of device configuration can be assessed consistently in terms of both, experimentally inaccessible internal processes, and measurable device characteristics. Apart from the obvious relevance for the design and analysis of ultra-thin and nanostructured absorber solar cell devices, the approach will also be valuable for the assessment of near-field thermophotovoltaic architectures [45, 46] and opto-electronic devices based on 2D materials and van der Waals heterostructures [47, 48] where similar challenges are encountered.

2 Theoretical approach: NEGF framework

The Non-Equilibrium Green’s Function (NEGF) formalism [49,50,51] used for this work considers electronic and optical modes in open systems under electronic and optical excitation (bias voltage and illumination) and provides electronic (scattering) rates and currents as well as optical rates (absorption, emission) and photon fluxes, which are evaluated consistently and on equal footing as derived directly from the NEGFs. The core of the formalism is the coupled solution of the equations for Green’s functions (G, \({\mathcal {D}}_{\gamma }\)) and self-energies (\(\varSigma\), \(\varPi _{\gamma }\)) for charge carriers and photons, respectively, as depicted in Fig. 1a. The NEGF framework for the microscopic description of opto-electronic processes in nanostructure-based solar cells has been presented in recent reviews (see, e.g., Ref. [52, 53]). Consideration of photonic modes for the primary photogeneration (rate \({\mathcal {G}}^{(0)}\)) in thin-film absorbers due to external illumination is discussed in [54, 55], and application to the assessment of reciprocity in ultra-thin absorber photovoltaics is shown in [38] for the case of normal incidence/emission. The assessment of the \(\hbox {V}_{\text {OC}}\) enhancement due to photon recycling, on the other hand, requires implementation of the full computational cycle shown in Fig. 1a. Below, we provide the explicit expressions of the quantities for which numerical results will be discussed, while further details on the theoretical and computational aspects of the formalism are given in the Supplementary Information (SI) and in the references.

Fig. 1
figure 1

a Schematic representation of the self-consistent opto-electronic NEGF formalism used to assess photon recycling in ultra-thin absorber devices. The coupling between electronic and optical degrees of freedom proceeds via the electron-hole polarization function and the electron–photon self-energy. b Sketch of the layer stack composing the ultra-thin absorber solar cell. The electrical simulation is restricted to the GaAs layers, with the boundaries of the doped regions as contacts

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2.1 Electronic characteristics

The electronic quantities are formulated for a two band model of a semiconductorFootnote 1, with valence (v) and conduction (c) bands coupled only via radiative transitions. The general expression for the particle current in terms of the charge carrier Green’s functions \(G_{b}\) (\(b=c,v\)) reads

$$\begin{aligned} J_{b}(z)=&\mp \lim _{z'\rightarrow z}\frac{e\hbar }{m_{0}}\big (\partial _{z}-\partial _{z'}\big ) \Big [\nonumber \\ {}&\times \frac{1}{{\mathcal {A}}}\sum _{{\mathbf {k}}_{\parallel }}\int \frac{{\mathrm d}E}{2\pi }G_{b}^{\lessgtr }({\mathbf {k}}_{\parallel },z,z',E)\Big ]. \end{aligned}$$
(1)

where \({\mathcal {A}}\) is the cross-sectional area and \({\mathbf {k}}_{\parallel }\) is the in-plane—or transverse—component of the charge carrier wave vector. The rate of primary charge carrier photogeneration due to external illumination with a (coherent) monochromatic electromagnetic wave of vector potential \({\mathbf {A}}\) and photon energy \(E_{\gamma }\) is given by

$$\begin{aligned} {\mathcal {G}}_{c}^{(0)}(z)=\sum _{{\mathbf {k}}_{\parallel }}\int \frac{{\mathrm d}E}{2\pi \hbar }\int {\mathrm d}z'\Big [&\varSigma _{c,e\gamma -\text {coh}}^{<}({\mathbf {k}}_{\parallel },z,z',E)\nonumber \\&\times G_{c}^{>}({\mathbf {k}}_{\parallel },z',z,E) \Big ], \end{aligned}$$
(2)

where the self-energy for the coupling to the coherent field reads [54]

$$\begin{aligned}&\varSigma _{c,e\gamma -\text {coh}}^{<}({\mathbf {k}}_{\parallel },z,z',E)=\frac{e^2}{m_{0}^2} \sum _{\mu }p_{cv}^{\mu }(z)p_{cv}^{\mu *}(z')\nonumber \\ {}&\quad \times G_{v}^{<}({\mathbf {k}}_{\parallel },z,z',E-E_{\gamma })A_{\mu }(\tilde{{\mathbf {q}}}_{\parallel },z,E_{\gamma }) A_{\mu }^{*}(\tilde{{\mathbf {q}}}_{\parallel },z',E_{\gamma }), \end{aligned}$$
(3)

where \(p_{cv}^{\mu }\), \(\mu \in \{x,y,z\}\) are interband momentum matrix elements and \(\tilde{{\mathbf {q}}}_{\parallel }\) is the in-plane photon wave vector of the incident field (i.e., zero for normal incidence). Quantification of the \(\hbox {V}_{\text {OC}}\)-enhancement due to photon recycling, on the other hand, requires the additional consideration of an electron–photon self-energy that couples the electronic GFs (\(G_{v,c}\)) to that of the electromagnetic field fluctuations (\(\varvec{{\mathcal {D}}}\)), describing the radiative recombination and secondary photogeneration processes associated with the incoherent emission of photons and their re-absorption, e.g., for electrons in the CB:

$$\begin{aligned} {\mathcal {R}}_{c,\text {rec}}(z)=&\sum _{{\mathbf {k}}_{\parallel }}\int \frac{{\mathrm d}E}{2\pi \hbar }\int {\mathrm d}z'\Big [\varSigma _{c,e\gamma -\text {inc}}^{>}({\mathbf {k}}_{\parallel },z,z',E)\nonumber \\ {}&\times G_{c}^{<}({\mathbf {k}}_{\parallel },z',z,E) \Big ], \end{aligned}$$
(4)
$$\begin{aligned} {\mathcal {G}}_{c}^{(1)}(z)=&\sum _{{\mathbf {k}}_{\parallel }}\int \frac{{\mathrm d}E}{2\pi \hbar }\int {\mathrm d}z'\Big [\varSigma _{c,e\gamma -\text {inc}}^{<}({\mathbf {k}}_{\parallel },z,z',E)\nonumber \\ {}&\times G_{c}^{>}({\mathbf {k}}_{\parallel },z',z,E) \Big ], \end{aligned}$$
(5)

where

$$\begin{aligned}&\varSigma _{c,e\gamma -\text {inc}}^{\lessgtr }({\mathbf {k}}_{\parallel },z,z',E)=~\frac{i\mu _{0}e^2}{m_{0}^2} \sum _{\mu }p^{\mu }_{cv}(z)p_{cv}^{\mu *}(z')\nonumber \\&\quad \times \int \frac{{\mathrm d}^{2}q_{\parallel }}{4\pi ^2}\int \frac{{\mathrm d}E_{\gamma }}{2\pi }~\Big \{ G_{v}^{\lessgtr }({\mathbf {k}}_{\parallel }-{\mathbf {q}}_{\parallel },z,z',E- E_{\gamma })\nonumber \\&\quad \times {\mathcal {D}}^{\lessgtr }_{\mu \mu } ({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\Big \}. \end{aligned}$$
(6)

2.2 Photonic characteristics

On the photon side, light–matter interaction is encoded in the photon self-energy \(\varvec{\varPi }\). The propagation and attenuation of the external coherent field \({\mathbf {A}}\) can be obtained from a standard transfer matrix method [56] with an extinction coefficient which is related to the photon self-energy via [57]

$$\begin{aligned} \kappa _{\mu }({\mathbf {q}}_{\parallel },z,E)=&\frac{(\hbar c_{0})^{2}}{4 n_{r}E^{2}}\int {\mathrm d}z' \mathfrak {R}\big [i\varPi _{\mu \mu }^{>}({\mathbf {q}}_{\parallel },z',z,E)\big ]. \end{aligned}$$
(7)

Since \(q_{\parallel }\ll k_{\parallel }\) the dependence of the photon SE on in-plane photon momentum is usually weak enough to be neglected. The spectral rates for incoherent emission and re-absorption of photons are expressed as follows [58]:

$$\begin{aligned} {\mathcal {R}}_{\text {reabs/em}}(z,E_{\gamma })=&-\int {\mathrm d}z'\sum _{\mu ,\nu }\int \frac{{\mathrm d}^2q_{\parallel }}{(2\pi )^2}\Big [{\mathcal {D}}^{\lessgtr }_{\mu \nu }({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\nonumber \\&\times \varPi ^{\gtrless }_{\nu \mu }({\mathbf {q}}_{\parallel },z',z,E_{\gamma })\Big ]/ (2\pi \hbar ) \end{aligned}$$
(8)

where the upper (lower) symbols relate to absorption (emission). The local photon flux normal to the slab surface is encoded in the non-equilibrium quantum statistical version of the Poynting vector as follows:

$$\begin{aligned} S_{z}(z,E_{\gamma })=&\int \frac{{\mathrm d}^2q_{\parallel }}{(2\pi )^2}s_{z}({\mathbf {q}}_{\parallel },z,E_{\gamma }), \end{aligned}$$
(9)
$$\begin{aligned} s_{z}({\mathbf {q}}_{\parallel },z,E_{\gamma })=&-\frac{E_{\gamma }}{4\pi \hbar }\lim _{z'\rightarrow z}\Bigg \{\partial _{z'}\sum _{\mu =x,y}\mathfrak {R}\Big [{\mathcal {D}}_{\mu \mu }^{>}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\nonumber \\&+ {\mathcal {D}}_{\mu \mu }^{<}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\Big ]-iq_{\parallel } \Big [{\mathcal {D}}_{xz}^{>}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\nonumber \\ {}&+ {\mathcal {D}}_{xz}^{<}({\mathbf {q}}_{\parallel },z,z',E_{\gamma }) +{\mathcal {D}}_{zx}^{>}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\nonumber \\ {}&+ {\mathcal {D}}_{zx}^{<}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\Big ]\Bigg \}, \end{aligned}$$
(10)

where the term proportional to \({\mathcal {D}}_{yy}\) corresponds to the transverse electric (TE) component and the remaining terms form the transverse magnetic (TM) component. Similarly, the modal slab absorptance for incident light of given polarization can be expressed as follows [59]:

$$\begin{aligned} a_{\text {TE}}({\mathbf {q}}_{\parallel },E_{\gamma })=&-\frac{1}{2}\int {\mathrm d}z\int {\mathrm d}z'\Big [\hat{{\mathcal {D}}}_{\text {vac},yy}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\nonumber \\&\times {\hat{\varPi }}_{yy}({\mathbf {q}}_{\parallel },z',z,E_{\gamma })\Big ], \end{aligned}$$
(11)
$$\begin{aligned} a_{\text {TM}}({\mathbf {q}}_{\parallel },E_{\gamma }) =&-\frac{1}{2}\int {\mathrm d}z\int {\mathrm d}z'\sum _{\mu =x,z}\Big [\hat{{\mathcal {D}}}_{\text {vac},\mu \mu } ({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\nonumber \\ {}&\times {\hat{\varPi }}_{\mu \mu }({\mathbf {q}}_{\parallel },z',z,E_{\gamma })\Big ], \end{aligned}$$
(12)

where \({\mathcal {D}}_{\text {vac}}\) is the vacuum-induced GF [59, 60], \(\hat{{\mathcal {D}}}\equiv {\mathcal {D}}^{>}-{\mathcal {D}}^{<}\) and \({\hat{\varPi }}\equiv \varPi ^{>}-\varPi ^{<}\). Finally, the generalized Kirchhoff law for the angle-integrated emitted photon flux based on the modal absorptance reads

$$\begin{aligned} \varPhi _{\text {em}}^{\lambda }(E_{\gamma })=&\int _{0}^{q_{0}} \frac{\mathrm {d}q_{\parallel }}{2\pi }q_{\parallel }~a_{\lambda }({\mathbf {q}}_{\parallel }, E_{\gamma })f_{\text {BE}}(E_{\gamma }-\varDelta \mu _{\text {cv}})/(\pi \hbar ), \nonumber \\ {}&\lambda =\text {TE,~TM}. \end{aligned}$$
(13)

2.3 Uniform DOS limit: connection to the Van Roosbroeck–Shockley theory

On the lowest level of approximation, the absorber/emitter is treated as an optically uniform medium, for which the GF of free field modes applies:

$$\begin{aligned} \bar{{\mathcal {D}}}_{0}^{>}(z,z',E_{\gamma })&=\sum _{\mu }\int \frac{\mathrm {d}^{2}q_{\parallel }}{(2\pi )^2}{\mathcal {D}}_{0,\mu \mu }^{>} ({\mathbf {q}}_{\parallel },z,z',E_{\gamma }) \end{aligned}$$
(14)
$$\begin{aligned}&=\left\{ \begin{array}{cc} -i q_{0}/\pi &{} \quad z=z',\\ -i\frac{\sin \{q_{0}(z-z')\}}{\pi (z-z')} &{}\quad \text {otherwise}, \end{array}\right. \end{aligned}$$
(15)

with \(q_{0}=n_{r}E_{\gamma }/(\hbar c_{0})\). Inserting this into the general expression for the emission rate (8) and neglecting the weak spatial dependence of the photon GF on the one hand and the momentum dependence of the photon self-energy on the other hand result in

$$\begin{aligned} {\mathcal {R}}_{\text {em}}^{(0)}(z,E_{\gamma })&\approx -\int \mathrm {d}z'\Big [ ~{\bar{\varPi }}^{<}({\mathbf {0}},z',z,E_{\gamma })\nonumber \\ {}&\quad \times \sum _{\mu }\int \frac{\mathrm {d}^{2}q_{\parallel }}{(2\pi )^2}{\mathcal {D}}_{0,\mu \mu }^{>}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })\Big ] \end{aligned}$$
(16)
$$\begin{aligned}&\approx -\frac{q_{0}}{\pi ^2\hbar }f_{\text {BE}}[E_{\gamma }-\varDelta \mu _{\text {cv}}(z)]\nonumber \\&\quad \times \int \mathrm {d}z'\mathfrak {I}\big [{\bar{\varPi }}^{R}({\mathbf {0}},z',z,E_{\gamma })\big ] \end{aligned}$$
(17)

where in addition, isotropy of the momentum matrix elements in the form of \(\varPi _{\mu \nu }={\bar{\varPi }}\delta _{\mu \nu }\) was assumed and the quasi-equilibrium approximation or Kubo–Martin–Schwinger relation (KMS) [61] \(\varPi ^{<}(E)f_{\text {BE}}(E-\varDelta \mu ){\hat{\varPi }}(E)\) and \({\hat{\varPi }}=2i \mathfrak {I}\big [\varPi ^R\big ]\) were used for the photon self-energy component. This is to be compared to the Van Roosbroeck–Shockley expression [20]

$$\begin{aligned} {\mathcal {R}}_{\text {em}}^{\text {VRS}}(z,E_{\gamma })=\alpha (z,E_{\gamma }) \frac{n_{r}^2(z,E_{\gamma })E_{\gamma }^2}{\pi ^2\hbar ^3c_{0}^2}f_{\text {BE}}[E_{\gamma }-\varDelta \mu (z)] \end{aligned}$$
(18)

by inserting the isotropic absorption coefficient derived from the photon self-energy:

$$\begin{aligned} \alpha (z,E_{\gamma })\approx -\frac{\hbar c_{0}}{n_{r}(z,E_{\gamma })E_{\gamma }}\int \mathrm {d}z' \mathfrak {I}\big [{\bar{\varPi }}^{R}({\mathbf {0}},z',z,E_{\gamma })\big ]. \end{aligned}$$
(19)

The resulting expression is indeed identical to the emission rate (17) if the explicit form of \(q_{0}\) is inserted in the latter. Hence, use of the isotropic free-field photon GF (14) in the electron–photon scattering self-energy reproduces the semiclassical result for the local radiative recombination rate as provided by the VRS detailed-balance approach.

2.4 Slab modes, non-local coupling and arbitrary non-equilibrium polarization in consistency with charge transport

Going beyond free-field modes requires solution of the Dyson and Keldysh equations for the retarded GF and the photon correlation functions (Eqs. (S17) and (S18) given in the SI). The NEGF problem of optical modes for slab geometry has been analyzed before in a range of investigations [59,60,61,62,63,64,65], however, explicit solutions were performed only for TE modes or neglecting spatial dispersion. Here, we obtain the full slab photon GF using a Dyson equation approach similar to the procedure used to evaluate the classical electromagnetic GF [66]. In order to describe optical rates and charge currents consistently and on the basis of identical electronic structure information, the optical response needs to be obtained in a microscopic picture of the charge carrier dynamics. On this most rigorous level, electron–photon coupling is considered via the non-local photon self-energy

$$\begin{aligned} \varPi _{\mu \nu }^{\alpha }({\mathbf {q}}_{\parallel },z,z',E)=&-\frac{i\hbar \mu _{0}e^2}{m_{0}^2}p^{\mu *}_{cv}(z) p_{cv}^{\mu }(z'){\mathcal {P}}_{cv}^{\alpha }({\mathbf {q}}_{\parallel },z,z',E)\nonumber \\&\times \delta _{\mu \nu },~ \alpha \in \{<,>,R\}, \end{aligned}$$
(20)

where \({\mathcal {P}}\) is the interband polarization function with components obtained directly from the electronic Green’s functions via non-equilibrium many-body perturbation theory [62, 67, 68]:

$$\begin{aligned} {\mathcal {P}}_{cv}^{\gtrless }({\mathbf {q}}_{\parallel },z,z',E_{\gamma }) =&\frac{1}{{\mathcal {A}}}\sum _{{\mathbf {k}}_{\parallel }}\int \frac{\mathrm {d}E}{2\pi \hbar }G_{c}^{\gtrless }({\mathbf {k}}_{\parallel },z,z',E_{\gamma })\nonumber \\&\times G_{v}^{\lessgtr }({\mathbf {k}}_{\parallel } -{\mathbf {q}}_{\parallel },z',z,E-E_{\gamma }), \end{aligned}$$
(21)
$$\begin{aligned} {\mathcal {P}}_{cv}^{R}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })=&\frac{1}{{\mathcal {A}}}\sum _{{\mathbf {k}}_{\parallel }}\int \frac{\mathrm {d}E}{2\pi \hbar }\Big [G_{c}^{R}({\mathbf {k}}_{\parallel },z,z',E)\nonumber \\&\times G_{v}^{<}({\mathbf {k}}_{\parallel } -{\mathbf {q}}_{\parallel },z',z,E-E_{\gamma })+G_{c}^{<}({\mathbf {k}}_{\parallel },z,z',E)\nonumber \\&\times G_{v}^{A}({\mathbf {k}}_{\parallel } -{\mathbf {q}}_{\parallel },z',z,E-E_{\gamma })\Big ]. \end{aligned}$$
(22)

Here, we use the electronic NEGFs that result from solution of the full quantum transport problem (Eqs. (S6) and (S7) in the SI) as indicated in Fig. 1a. The Keldysh equation for the correlation functions is thereby solved explicitly, i.e., without resorting to the quasi-equilibrium approximation. However, the classical limit—as discussed in [23]—is reproduced correctly, i.e., insertion of the quasi-equilibrium approximation of the Keldysh equation with a local continuum self-energy

$$\begin{aligned} \varPi _{\mu \nu }^{R}({\mathbf {q}}_{\parallel },z,z',E_{\gamma })=\frac{E_{\gamma }^{2}}{(\hbar c_{0})^2}\big [\varepsilon (z,E_{\gamma })-1\big ]\delta _{\mu \nu }\delta (z-z') \end{aligned}$$
(23)

defined in terms of the complex dielectric function \(\varepsilon\) yields the classical Poynting vector.

3 Numerical simulation results

Fig. 2
figure 2

Monochromatic current–voltage characteristics of a 50 nm GaAs p-i-n solar cell for a device a with a gold back reflector, and b without metallic reflector, computed with PR (full symbols) and without PR (empty symbols) as well as using the free-field photon DOS for the spontaneous emission (dashed lines). The peculiar shape of the curve around the maximum power point is due to the voltage dependence of the absorption coefficient (Franz–Keldysh effect)

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The full NEGF simulation approach is applied to ultra-thin GaAs p-i-n solar cells with carrier selective contacts, an absorber thickness of 50 nm and featuring either no reflector or a gold mirror, which is either attached to the hole contact or separated by a buffer layer with electrical and/or optical functionality (Fig. 1b). Details on the electronic structure model used, the material parameters and the numerical settings are given in the SI.

The ability to assess the impact of photon recycling on the electrical characteristics of thin-film solar cells is demonstrated in Fig. 2, which shows the current–voltage characteristics under monochromatic illumination with photon energy \(E_{\gamma }=1.44\) eV and intensity \(I_{\gamma }=0.1\) W/\(\hbox {cm}^{2}\),Footnote 2 in (a) presence and (b) absence of a metallic reflector, with PR (full symbols) and without PR (empty symbols, full lines). For comparison, the characteristics without PR obtained by using the uniform photon DOS in the local emission rate (VRS approximation) are shown as well (dashed line). While the renormalized photon DOS characteristics of the device with metallic reflector show a reduction in open circuit voltage with respect to the free-field case even in the presence of photon recycling, there is a pronounced \(V_{\text {OC}}\) increment due to re-absorption observed for the device without reflector. The peculiar enhancement of the photocurrent around the maximum power point is related to an increase of the absorption coefficient with decreasing field at larger voltage (Franz–Keldysh effect [36,37,38,39]), which is amplified by the voltage increment from photon recycling.

To investigate the origin of the non-intuitive voltage loss in the device with reflector—that on the other hand yields a much higher photocurrent due to larger absorptance caused by the proximity to the first Fabry–Perot resonance at around 30 nm slab thickness—and of the sizable boost in PR-related \(V_{\text {OC}}\) enhancement from \(\varDelta V_{\text {OC}}\approx\) 20 mV in the device with reflector to \(\varDelta V_{\text {OC}}\approx\) 80 mV in the one without, the energy-integrated local volume rates of primary photogeneration \({\mathcal {G}}\) due to external illumination, radiative recombination \({\mathcal {R}}_{\text {em}}\) due to internal emission and secondary photogeneration \({\mathcal {R}}_{\text {reabs}}\) due to re-absorption of internally emitted photons are evaluated. As shown in Fig. 3, which displays the optical rates at an external voltage of V = 1.05 V (dashed red lines in Fig. 2), all of the rates are strongly enhanced in the system with attached metallic reflector as compared to both the slab in air under consideration of the actual photon DOS, and the treatment of the emission based on free-field modes. For the latter, the result found by using the full NEGF formalism with the quasi-equilibrium approximation of the free-field GF (dashed) is in perfect agreement with the predictions of the analytical VRS expression (18) (symbols) based on the (NEGF) absorption coefficient, the black-body photon flux and the local quasi-Fermi level splitting (for which the external voltage V is used: \(\varDelta \mu =\mathrm{eV}\)). The smoothness of the rates is due to quantum mechanical suppression of the electronic local DOS close to the carrier selective contacts [36]. In contrast to the desired increase in the primary photogeneration rate (solid lines) which has its maximum close to the center of the absorber, the rates of emission (dashed lines) and re-absorption (dash-dotted lines) in the reflector case show a strong enhancement in the vicinity of the mirror. The disparity in \(\varDelta V_{\text {OC}}\) can directly be traced back to the much lower ratio of re-absorption to internal emission in the presence of the metallic mirror.

Fig. 3
figure 3

Local volume rates of coherent photogeneration \({\mathcal {G}}\) (full lines), incoherent spontaneous emission \({\mathcal {R}}_{\text {em}}\) (dashed lines), and re-absorption \({\mathcal {R}}_{\text {reabs}}\) (dash-dotted lines), for the system with gold reflector (ocher) and without reflector (green) at an external voltage \(V=1.05\) V (red lines in Fig. 2). The emission and re-absorption rates display a large enhancement in the vicinity of the mirror. The disparity in \(V_{\text {OC}}\) enhancement observed in Fig. 2 can be related directly to the much lower ratio of re-absorption to internal emission in the presence of the metallic reflector. For the case of a uniform density of photon states (magenta), the emission rate \({\mathcal {R}}_{\text {em}}^{(0)}\) based on the free-field GF is compared to the VRS rate (Eq. 18) obtained from the absorption coefficient and the QFLS \(\varDelta \mu =\mathrm{eV}\) (symbols) (Color figure online)

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Analysis of the modal structure of the emission rate (at \(V=1.05\) V) in terms of the position-integrated spectral rate as a function of in-plane photon momentum \(q_{\parallel }\) and photon energy \(E_{\gamma }\) identifies emission into guided modes as the dominant radiative loss channel in the absence of a metallic reflector (Fig. 4a); in the case of the ultra-thin absorber, only the fundamental transverse electric mode is present, whose analytical dispersion as obtained from the standard solution of the transcendental equation [69] is displayed as a green dashed line. In the device with metallic reflector, on the other hand, the radiative loss is dominated by hybridization of the guided modes with the surface plasmon polariton (SPP) resonance at the GaAs–Au interface (Fig. 4b). The green dashed line for the SPP mode shows the approximate analytical SPP dispersion based on the dielectric functions of GaAs and Au,

$$\begin{aligned} q_{\parallel }(E_{\gamma })=\frac{E_{\gamma }}{\hbar c_{0}}\left[ \frac{\varepsilon '_{\text {GaAs}}(E_{\gamma })\varepsilon '_{\text {Au}}(E_{\gamma })}{\varepsilon '_{\text {GaAs}}(E_{\gamma })+\varepsilon '_{\text {Au}}(E_{\gamma })}\right] ^{1/2}, \end{aligned}$$
(24)

where \(\varepsilon '\) denotes the real part of the dielectric function [32].

Fig. 4
figure 4

Position-integrated emission rate vs. in-plane photon momentum and photon energy (\(V=1.05\) V): a The radiative recombination rate in the slab bounded by air is dominated by emission into guided (TE) modes (dashed lines: analytical dispersions of the fundamental TE/TM wave guide modes). b In the presence of a gold reflector, it is guided modes that are hybridized with surface plasmon polaritons that contribute the main radiative loss channel (dashed line for SPP: analytical GaAs-Au surface plasmon polariton dispersion)

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Fig. 5
figure 5

Total internal emission rate at \(V=1.05\) V integrated spatially over the active absorber layer (solid purple line), externally emitted photon flux (dotted dark green line) and spatially integrated net internal emission rate (dashed ocher line), where the re-absorption is accounted for: a In the case of the device without metallic reflector, all of the internally emitted light is either recycled (purple area) or coupled out (dark green area). b In the presence of a metallic reflector, only a part of the internal emission that lies outside of the loss cone is recycled, while the remaining part (ocher area) is lost due to parasitic absorption (Color figure online)

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Fig. 6
figure 6

Impact of built-in field and contact configuration on the local volume rates of generation, emission and re-absorption: a Conduction band edge profiles for different doping densities \(N_{\text {d}}\) at \(V=1.1\) V and valence band edges for different buffer layers at the hole contact for \(V=1.05\) V. b Absorptance and emitted photon flux of the system without metallic mirror at \(V=1.1\) V, reflecting the effect of the built-in field. In both high and low field conditions, the two quantities can be related by the generalized Kirchhoff law. The red arrow marks the photon energy of the monochromatic external illumination. c Local rates for low- and high-field conditions at \(V=1.1\) V, revealing a detrimental impact of large doping concentrations. d Local rates for different contact configurations (\(V=1.05\) V): no buffer, 5 nm/10 nm hole blocking layer (index matched), 5-nm HBL with \(n_{\text {r}}=2\), and a 5-nm air gap (detached reflector). The electronic passivation via the HBL reduces all of the rates together, while the dielectric passivation keeps the level of primary generation, but reduces emission and re-absorption down to the level of the device without metallic reflector

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The effect of parasitic absorption on the effectiveness of photon recycling can be quantified further by comparing the spatial integration of the net internal emission rate

$$\begin{aligned} \phi _{\text {em}}^{\text {int,net}}(E_{\gamma })=\int {\mathrm d}z \left[ {\mathcal {R}}_{\text {em}}(z,E_{\gamma })-{\mathcal {R}}_{\text {reabs}}(z,E_{\gamma })\right] \end{aligned}$$
(25)

with the total flux due to internal emission

$$\begin{aligned} \phi _{\text {em}}^{\text {int,total}}(E_{\gamma })=\int {\mathrm d}z ~{\mathcal {R}}_{\text {em}}(z,E_{\gamma }) \end{aligned}$$
(26)

and with the externally emitted photon flux

$$\begin{aligned} \phi _{\text {em}}^{\text {ext}}(E_{\gamma })=\left[ S_{z}(z_{R},E_{\gamma })-S_{z}(z_{L},E_{\gamma })\right] /E_{\gamma }, \end{aligned}$$
(27)

where \(z_{L,R}\) denote the left and right boundaries of the active absorber layer, respectively. These quantities are shown in Fig. 5a and b for the devices without and with metallic reflector, respectively. In the absence of a metallic reflector, external flux (dotted green line) and net internal emission (dashed ocher line) coincide, i.e., all of the internally emitted photons (solid purple line) are either recycled (light purple area) or coupled out (light green area). In the presence of a metallic mirror, on the other hand, only a fraction of the internal emission outside the loss cone is recycled, while the rest is absorbed parasitically (light ocher area). This also means that in this case, the generalized Kirchhoff law [21] does no longer provide a valid optical limit of the current voltage characteristics with photon recycling [4, 70].

In order to shed light on the interplay of optical and electronic degrees of freedom at the nanoscale and relate them to device parameters, we proceed by quantifying the effects of built-in fields and contact configuration on the optical rates inside the absorber. The upper part of Fig. 6a displays the conduction band profile of the reflector-free case at an external voltage \(V=1.1\) V for different levels of doping density \(N_{\text {d}}\) considered in the solution of Poisson’s equation that is coupled self-consistently to the electronic NEGF transport problem, resulting in strongly varying built-in fields, from a close to field free flat band situation at \(N_{\text {d}}=3\times 10^{15}\) \(\hbox {cm}^{-3}\) (light solid line) to a pronounced band bending at \(N_{\text {d}}=10^{18}\) \(\hbox {cm}^{-3}\) (dark solid line). For comparison, the band profile is also shown for the default doping density of \(N_{\text {d}}=2\times 10^{16}\) \(\hbox {cm}^{-3}\) used in the rest of the simulations (dashed line). The high-field characteristics of the absorptance at normal incidence and \(V=1.1\) V (Fig. 6b, dark solid line) exhibit the features of the Franz–Keldysh effect as observed experimentally [39]. The field-induced band tailing results in a pronounced broadening of the emitted photon flux—for which the angular integration of the z-component of the NEGF Poynting vector at the slab surfaces is shown (dashed lines)—as compared to the flat band characteristics (light solid/dashed lines). The validity of the generalized Kirchhoff relation between absorptance and emitted flux (symbols) is verified for the entire angular range, hence beyond Ref. [38] that was restricted to normal incidence and perpendicular emission. Figure 6b indicates a larger overlap between absorption and emission in the flat band case. However, the corresponding local rates displayed in Fig. 6c (solid line: generation, dashed line: emission, dash-dotted line: re-absorption) reveal that while there is a slightly larger ratio of re-absorption to emission, it is the field-induced increase in the emission rate that dominates the characteristics. Finally, the impact of an electronic and/or optical passivation layer is investigated for the device architecture with metallic reflector. As shown in Fig. 6d, introduction of a semiconducting hole blocking layer (HBL, green: 5 nm, magenta: 10 nm; valence band profile at \(V=1.05\) V shown in the lower part of Fig. 6a) with the refractive index of GaAs leads only to a moderate reduction of the plasmonic enhancement of emission and re-absorption, but is accompanied by an undesired decrease of the primary photogeneration. This electronic passivation effect is based on a reduction of the electron–hole polarization close to the contact due to suppression of the hole density and hence increases with the thickness of the HBL. Combination with dielectric passivation via lowering of the refractive index of the 5 nm HBL to \(n_{\text {r}}=2\) (ocher)— similar to that of standard transparent conductive oxides—yields a much more pronounced decrease in the radiative recombination and more favorable relation to re-absorption (i.e., less parasitic losses) at even slightly increased primary photogeneration. The reason fo this behavior is that, for an optical buffer with lower refractive index, the plasmon peak is shifted to lower values of transverse wave vector, which reduces its weight in the integration over transverse photon wave vector in the expressions for the optical rates (Eq. 8). Detaching the reflector by inserting a 5-nm air gap (red) lowers the emission and re-absorption toward the levels observed for the architecture without reflector, while the primary photogeneration rate is maintained close to the high value of the system without any buffer layer (light blue). A similar observation regarding the beneficial impact on PR by detaching the reflector was recently made in the context of near-field thermophotovoltaic applications [46].

Fig. 7
figure 7

Current–voltage characteristics of the ultra-thin GaAs solar cell with different reflector configurations: a without reflector (purple squares), b with a gold reflector (green circles), c with a detached reflector (light blue upward triangles), d with an index-matched HBL (gold inverted triangles), and e with an HBL of increased refractive index \(n_{\text {r}}=2\) (red diamonds). For comparison, the characteristics without PR are indicated by dashed lines. For the HBL configurations, also shown are the results for contact selectivity restricted to electrons (with PR: dashed-dotted lines, w/o PR: dotted lines). The opto-electronic decoupling of the metallic reflector mitigates a large part of the losses associated with plasmon-related parasitic absorption, and full contact selectivity is instrumental to benefit from PR (Color figure online)

Full size image

Figure 7 displays the current–voltage characteristics of the ultra-thin GaAs solar with the different contact configurations for which the rates are shown in Fig. 6d, both with PR (full lines with symbols) and without PR (dashed lines). As shown in Fig. 7c, dielectric passivation with an optical buffer (detached reflector) in combination with perfect contact selectivity enables device characteristics with up to \(\sim\)80% of the short circuit current of the system with reflector and a \(V_{\text {OC}}\) decrement of only \(\sim\)20 mV from the device without reflector (Fig. 7a). The contribution of photon recycling is thereby reduced to \(\varDelta V_{\text {OC}}\approx\) 40 mV, which is still much larger than the value found in the absence of dielectric passivation (Fig. 7b). However, as shown in Fig.  7d, consideration of the actual incomplete selectivity due to finite height of the hole extraction barrier at the 5 nm HBL degrades \(V_{\text {OC}}\) via leakage contributions to dark current to an extent that basically prevents any benefit from photon recycling [with PR: dash-dotted lines, no PR: dotted lines—shown only for (d) and (e)]. Hence, dielectric passivation via low-index buffer layers (Fig. 7e) improves device performance only as long as carrier selectivity is close to ideal.

4 Conclusions

We demonstrate the potential of fully coupled electronic and photonic NEGF simulation of nanoscale opto-electronic devices for the rigorous assessment of photon recycling in ultra-thin absorber solar cells where both optical and electronic properties are affected by the effects of finite size. The treatment enables evaluation of \({V}_{\text {OC}}\) enhancement directly from the electrical characteristics, i.e., under consideration of both, optical cavity effects and the full transport problem including contact layers, for which macroscopic approaches are not appropriate. The appearance of, on the one side, massive \(\varDelta V_{\text {OC}}\) in device architectures where plasmonic effects are absent or attenuated by electronic and/or dielectric buffer layers, and, on the other side, the effects of bias-dependent built-in fields on the level of generation, emission, and re-absorption underline the merits of a comprehensive quantum simulation approach at the limit of ultra-scaled devices where optical and electronic properties are interrelated and intimately connected to the geometrical device configuration.