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Photonic Design: From Fundamental Solar Cell Physics to Computational Inverse Design

Owen D. Miller

arXiv:1308.0212v1physics.optics

TL;DR

The dissertation addresses how to approach high solar-cell efficiency limits and how to move beyond intuition-limited photonic design. It develops photon-management analysis and a shape-calculus inverse-design framework, reporting a 33.5% theoretical GaAs efficiency limit and applications to optical cloaking and thin-film solar cells. The work also shows that realistic luminescence and computational cost constrain these approaches.

  • Problem

    Existing analyses and design practices do not fully account for realistic luminescence and optical extraction, while conventional photonic design restricts structures to engineer-intuited possibilities.

  • Method

    The dissertation extends quasi-equilibrium efficiency analysis to photon recycling and imperfect radiation properties, and uses shape calculus for computational inverse design.

  • Results

    33.5% is the theoretical efficiency limit calculated for GaAs, achievable for planar or textured fronts with a mirrored rear surface.

  • Takeaways & Limitations

    High voltages require maximizing external luminescence, while inverse design can efficiently compute non-intuitive photonic structures for specified responses.

  • Takeaways & Limitations

    The computational implementation scales as 6NxNyNz simulations for a grid with Ni cells along each dimension, making computational cost its primary drawback.

Abstract

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Photonic innovation is becoming ever more important in the modern world. Optical systems are dominating shorter and shorter communications distances, LED's are rapidly emerging for a variety of applications, and solar cells show potential to be a mainstream technology in the energy space. The need for novel, energy-efficient photonic and optoelectronic devices will only increase. This work unites fundamental physics and a novel computational inverse design approach towards such innovation. The first half of the dissertation is devoted to the physics of high-efficiency solar cells. As solar cells approach fundamental efficiency limits, their internal physics transforms. Photonic considerations, instead of electronic ones, are the key to reaching the highest voltages and efficiencies. Proper photon management led to Alta Device's recent dramatic increase of the solar cell efficiency record to 28.3%. Moreover, approaching the Shockley-Queisser limit for any solar cell technology will require light extraction to become a part of all future designs. The second half of the dissertation introduces inverse design as a new computational paradigm in photonics. An assortment of techniques (FDTD, FEM, etc.) have enabled quick and accurate simulation of the "forward problem" of finding fields for a given geometry. However, scientists and engineers are typically more interested in the inverse problem: for a desired functionality, what geometry is needed? Answering this question breaks from the emphasis on the forward problem and forges a new path in computational photonics. The framework of shape calculus enables one to quickly find superior, non-intuitive designs. Novel designs for optical cloaking and sub-wavelength solar cell applications are presented.

Introduction

The dissertation connects solar-cell physics with computational photonic design. It argues that high-efficiency solar cells should be designed for external emission and introduces inverse design to compute structures for desired electromagnetic responses.

  • Solar-cell physics: Solar cells should be designed as efficient light emitters because maximizing external emission increases voltage and efficiency.This reframes external emission from a loss mechanism into a design objective for approaching ideal limits.
  • Computational inverse design: Traditional photonic design relies on researcher intuition, restricting the explored design space to what the engineer can imagine.Forward solvers compute the response of a specified structure, but do not directly solve for the structure producing a desired response.
  • Computational inverse design: The dissertation introduces shape-calculus-based inverse design, which searches for structures that provide desired responses and can compute non-intuitive designs efficiently.The framework is applied to optical cloak design and thin-film solar cells operating at sub-wavelength scales.
  • Solar-cell physics: Solar-cell efficiency matters economically because installation and related costs scale with module area, making high-efficiency cells important for cost parity.The passage contrasts 10% efficient modules at about $0.06/kWh with 25% efficient modules costing about 300$/m2 at the same electricity cost.
  • Solar-cell physics: Shockley-Queisser limits can be highly sensitive to small imperfections, such as rear-mirror reflectivity and material or geometric quality.A 98% reflective rear mirror provides double the external emission of a 96% reflective mirror.

Luminescent Extraction Determines the Voltage

The chapter shows that open-circuit voltage, and therefore operating-point voltage and fill factor, is primarily controlled by external luminescence. Because extraction losses accumulate through re-absorption, non-radiative recombination, and imperfect interfaces, solar-cell geometry and photon management determine voltage and efficiency.

  • Voltage and operating point: Open-circuit voltage primarily determines operating-point voltage and fill factor, while current is set more directly by photon absorption and carrier extraction.Non-radiative recombination modifies the voltage dependence but does not remove open-circuit voltage as the prime determinant.
  • External luminescence: Poor external light extraction imposes a voltage loss because external yield ηext is at most one.At open circuit, internally recombining carriers contribute to voltage according to how many emitted photons escape externally.
  • Entropic penalties: The largest entropic voltage penalty is about −10.7kT from the mismatch between solar and emission solid angles, while later terms contribute slightly less than 1kT.The external-yield contribution is highly variable and therefore remains a key device-dependent term.
  • Voltage limit: For bandgaps from 1–1.8eV, the 260meV offset is within ±kT, and external luminescence yield determines proximity to the ideal open-circuit voltage.The 260meV value is almost exact for a 1.4eV bandgap.
  • Extraction difficulty: At ηint = 99% and optical thickness αL = 2.5, plane-parallel geometry yields only about 50% external yield because internal imperfections are amplified through repeated photon cycling.For internal yield below approximately 99%, external yield depends inversely on the small deviation from unity.
  • Geometry and photon management: Random surface texturing can raise external yield by a factor of about 10 over plane-parallel geometry, producing a voltage boost of approximately 60meV.The benefit is strongest when re-absorption does not sufficiently randomize photons and texturing enables full absorption at smaller thickness.

Approaching the Shockley-Queisser Efficiency Limit

Approaching the Shockley–Queisser limit requires treating external luminescence and photon extraction as central determinants of voltage and efficiency. For GaAs, careful optical design supports a theoretical efficiency of 33.5%, while material and optical losses constrain other systems.

  • At open circuit, efficient external luminescence indicates low optical and non-radiative losses because emitted photons balance absorbed sunlight and support carrier-density buildup.Additional non-radiative recombination or photon loss lowers carrier density and open-circuit voltage.
  • The failure to extract recycled internal photons efficiently indicates accumulated non-radiative losses that prevent the best solar cells from reaching the Shockley–Queisser limit.Repeated escape attempts through a narrow escape cone require internal luminescence efficiency far above 90%.
  • Light extraction must be designed into high-performance solar cells because the Shockley–Queisser limit cannot be achieved without minimizing non-radiative losses.Although extracted photons do not directly contribute to performance, near-100% external extraction at open circuit gauges low optical losses.
  • 99.7% internal luminescence yield has been experimentally measured in high-quality GaAs, but voltage depends on external luminescence, which also reflects optical design quality.Absorbing contacts or a faulty rear mirror remove photons that could otherwise be recycled.
  • 33.5% is the calculated theoretical maximum efficiency for a GaAs solar cell under the one-sun AM1.5G spectrum and measured GaAs absorption curve.This exceeds the 26.4% record cited for GaAs in 2010 and is theoretically achievable with either planar or textured fronts when the rear surface is mirrored.
  • A perfectly reflecting rear mirror can improve efficiency and voltage without increasing short-circuit current, because it redirects internal photons toward front-surface emission and enables photon recycling.The three studied geometries vary front-surface texturing and rear-mirror quality to evaluate these optical effects.

Analysis of next-generation solar cells

Next-generation solar-cell concepts are highly sensitive to luminescence and photon-extraction losses, which primarily reduce voltage rather than current. These penalties can sharply lower theoretical efficiencies, especially in low-bandgap and up-conversion systems.

  • Fundamental limits: 33% is the approximate single-junction efficiency ceiling imposed by sub-bandgap absorption, carrier thermalization, and required luminescence.The luminescence condition follows from thermodynamic detailed balance and should be maximized rather than avoided.
  • Luminescence losses: A voltage penalty of −kT ln(0.02) ≈100mV occurs at 2% luminescence yield, independently of bandgap, making small-bandgap cells relatively more vulnerable.Large-bandgap cells are comparatively robust to imperfect photon management.
  • Carrier multiplication: 90% internal luminescence yield can reduce carrier-multiplication efficiency by more than 7% at the optimal bandgap and about 12% at the smallest bandgap.At an 80% yield and 0.7eV bandgap, efficiency falls to 34.9%.
  • Up-conversion: 90% internal yield in up-conversion reduces maximum-concentration and minimum-emission efficiencies from 58.5% and 62.8% to below 40% and 30%, respectively.The calculation models non-ideal up-converter yield; additional solar-cell luminescence losses are not included.
  • Up-conversion: Up-conversion can return almost 30mA/cm2 to a 2eV solar cell, but its thermodynamically required luminescent emission is extremely susceptible to loss.The same emission process that provides sub-bandgap current also creates a strong sensitivity to extraction imperfections.
  • Up-conversion: Poor extraction is especially damaging in up-converters because their luminescence depends linearly on extraction efficiency, amplifying emission losses.Texturing raises external yield to approximately 15% from 6% in the plane-parallel geometry, but the improvement remains limited.

A New Photonic Inverse Design Method

Photonic inverse design reframes computational electromagnetics from predicting fields for fixed geometries to finding geometries that approximate desired responses. The proposed shape-calculus framework targets this inverse problem with substantially fewer simulations than stochastic alternatives.

  • Forward methods compute electromagnetic fields or eigen-frequencies for a given geometry, whereas inverse design seeks a geometry producing a desired response.The inverse problem is generally non-unique or may lack a realizable exact solution, so it is formulated as achieving the response as closely as possible.
  • PDE-constrained optimization maximizes a merit function of electromagnetic fields and frequency while enforcing Maxwell’s equations.The design variables are the permittivity and permeability tensors.
  • Tens of thousands of simulations usually make stochastic algorithms unrealistic for three-dimensional optimizations with sizeable parameter spaces.The simulation count is a primary efficiency constraint because solving the forward problem dominates computational cost.
  • Shape calculus requires only two simulations to compute the objective derivative over the entire infinite-dimensional space of shapes and topologies, with typically fewer than a hundred iterations to converge.It extends differential calculus to shape and topology and is presented as the mathematical basis of the new photonic design method.
  • The method is developed through an intuitive electromagnetic treatment of shape calculus, followed by implementation considerations and an analysis of Maxwell-equation symmetries.The framework is first introduced through a two-dimensional scalar example before extension to three-dimensional problems.

Simple example

In a simplified two-dimensional optimization, shape calculus evaluates how small dielectric changes affect a field-concentration merit function. Reciprocity converts the required information into direct and adjoint simulations that produce a gradient over the design domain.

  • The example maximizes electric-field intensity at x0 in a two-dimensional TE-polarized structure with scalar fields and material parameters.The setup assumes translational symmetry in the out-of-plane direction and an incident wave from above.
  • For a small inclusion, linearization makes the merit-function change depend on the induced dipole and the Green’s function connecting its location x′ to x0.Ignoring higher-order terms is valid for small geometrical changes and supports iterative optimization around the local geometry.
  • Brute-force optimization tests a separate inclusion at every candidate location, retains the one with the largest merit-function increase, and repeats the process.This approach requires one simulation for each possible x′ to evaluate the corresponding Green’s function.
  • Reciprocity gives G(x0, x′) = G(x′, x0), allowing the source and observation roles to be exchanged in the Green’s function.The exchanged form is interpreted as an adjoint dipole driven at x0 with amplitude Eold(x0).
  • Two simulations—a direct field solution and an adjoint dipole solution—provide the information needed to evaluate the shape gradient across all possible inclusion locations.The gradient is iterated by selecting the most promising inclusion and updating the geometry.
  • The derivation uses the continuous design problem, which differs slightly from the discretized problem except in the limit of infinite resolution.The simple example is also explicitly described as unrealistically simple, while arbitrary boundary movements can be approximated by induced dipoles.

General case

The general formulation extends the dipole-and-reciprocity argument to three-dimensional vector fields, tensor materials, and merit functions depending on fields across regions. The resulting adjoint formulation still determines the objective variation with two simulations.

  • The general case allows vector electromagnetic fields, tensor permittivity and permeability, and merit functions that depend on fields throughout a region.Eigen-frequency objectives are also possible but require substantially different treatment.
  • Shape and topological changes alter permittivity within a region ψ and induce a polarization density Pind that generates the field variations.The induced polarization is the common representation for both boundary changes and new inclusions.
  • Green’s-function tensors map induced polarization to electric and magnetic field variations, with electric dipoles retained for small geometric changes.Magnetic dipoles enter when required by the symmetry relations.
  • Reciprocity exchanges electric-dipole source and observation points and relates magnetic-field responses to electric fields from magnetic dipoles.These relations enable the field variation to be rewritten using an adjoint field.
  • Two simulations determine the objective variation for any geometric deformation: one supplies the steady-state fields and induced polarization, and the adjoint one supplies EA(x′).The merit-function derivatives determine the adjoint sources, after which δF is available throughout the design domain.
  • The general variation formula measures a selected deformation’s effect but must be further manipulated into a gradient for inverse-design use.The method separates topological deformations from shape deformations.

Topological Derivative

For a new dielectric inclusion, the topological derivative models the inclusion through its induced dipole and shape-dependent polarizability. Evaluating the resulting expression across the design domain yields an iterative placement rule.

  • Creating a dielectric inclusion of permittivity ϵ2 inside material with permittivity ϵ1 is treated as a topological change.The inclusion’s induced dipole is characterized using a shape-dependent polarizability α.
  • In the vanishing-inclusion limit, the fields in the derivative expression are approximated at the inclusion center x′, with the inclusion volume V retained.This produces a localized expression for the merit-function variation.
  • The algorithm simulates E and EA, evaluates the topological-derivative expression over the designable domain, inserts material where δF is maximal, and iterates.This converts the derivative into a direct rule for selecting each new inclusion.

Shape Derivative

Shape calculus computes boundary updates from field information gathered across the design region, using two simulations rather than exhaustive testing. Its level-set representation supports flexible deformation and merging of arbitrarily shaped boundaries.

  • Boundary deformation: Shape derivatives must use continuous electromagnetic-field quantities because electric-field components can be discontinuous across material boundaries.Maxwell boundary conditions preserve tangential E and normal D, not every component of E.
  • Boundary deformation: Boundary shape derivatives provide an update procedure requiring only two simulations.The right-hand side acts as an infinite-dimensional shape derivative evaluated over the boundary surface.
  • Information efficiency: Brute-force optimization wastes computation because each test simulation provides only one additional relevant field value at x′.The full domain is simulated even though fields at other locations do not contribute useful information for that test.
  • Information efficiency: The adjoint simulation makes fields at all N design points useful, yielding 2N information units with only about 2N simulated data points.This avoids the irrelevant information computed by the N−1 brute-force test simulations.
  • Level-set representation: Level-set methods represent an arbitrarily shaped boundary as the zero level set of a signed-distance function φ.Updating φ and recovering φ = 0 enables continuous deformation, merging, and separation without independently tracking boundary particles.

Optical Cloak Optimization

Inverse design uses shape calculus to optimize cloaks under material and functionality constraints rather than relying on transformation-optics prescriptions. The resulting designs trade material complexity and performance across angular, frequency, and material-realism requirements.

  • Transformation-optics limitations: Transformation optics requires simultaneous permittivity and permeability transformations that are generally anisotropic, exotic, and difficult to constrain practically.The transformed parameters can involve full nine-component matrices, negative values, or values tending to infinity.
  • Inverse-design framework: Inverse design produces reduced-material cloaks with reduced functionality while allowing designers to choose the tradeoffs.Real material parameters, including imperfect conductors, can be incorporated directly into the optimization.
  • Optimization results: Within thirty iterations, scattering width reached about 10^-2, nearly three orders of magnitude below its initial value.The optimized structure retained relatively low permittivity contrast but formed a complex pattern difficult to design intuitively or optimize stochastically.
  • Optimization results: Perfect rotational symmetry required anisotropic permittivity because isotropic permittivity was insufficient.Using anisotropic ε achieved a sufficiently small scattering width to be considered a cloak with complete angular independence.
  • Optimization results: For a four-fold-symmetry design, σ < 0.1 occurred over relative bandwidth Δω/ω0 ≈ 1/20 and incidence angles |θinc| < 2°.The broader thresholds σ < 1 corresponded to Δω/ω0 ≈ 1/6 and |θinc| < 8°.
  • Material realism: Using real gold parameters initially severely diminished cloaking, but reoptimization recovered cloaking with significantly reduced scattering width.The resulting design placed dielectric directly in front of the gold to redirect waves around the lossy material.

Surface Textures for Sub-Wavelength Solar Cells

The section develops inverse-designed surface textures for thin, high-index solar absorbers, targeting broadband light-trapping enhancement rather than necessarily exceeding the 4n^2 limit. The optimized structure achieves strong enhancement across energy, angle, and polarization while reducing computational cost through normal-incidence optimization.

  • Motivation: Light trapping is fundamental for solar cells, but high-index sub-wavelength structures require designs beyond previously proposed low-index modal enhancements.The goal is large enhancement in thin-film, high-index structures relevant to photovoltaic technology.
  • Problem formulation: The enhancement factor measures absorptivity relative to material absorption, while the merit function maximizes the minimum enhancement across a frequency band.Angle averaging is computationally expensive, so the optimization uses normal-incidence absorptivity and seeks broadband rather than highly resonant performance.
  • Optimization setup: The design optimizes a textured surface on a 150nm absorber with an anti-reflection coating, low-index surroundings, and a perfect rear reflector using transmission as an absorption proxy.The surface is parameterized with a smooth Fourier basis, and the front-monitor transmission avoids directly recording three-dimensional field data.
  • Optimization results: 78 is the average enhancement of the final structure, with enhancement greater than 50 across the stated energy range.The optimized surface has approximately 160nm peak-to-valley variation and no imposed symmetry beyond the coefficient relation required by a real-valued surface.
  • Robustness and cost: 40 is the angle- and polarization-averaged enhancement, while enhancement remains above 20 for every orientation despite an off-angle penalty.The result supports normal-incidence optimization because full angular and polarization optimization would increase runtime from 3 days to 75 days, about 25 times longer.
  • Implications: A factor of 40 approaches the ray-optics limit 4n^2 ≈50 and could reduce a currently 1µm-thick cell to 25nm.The structure also surpasses the calculated enhancement factor limit of 31 for a 150nm planar thin film, indicating that texturing contributes both coupling and additional functionality.

Appendix A

Appendix A develops reciprocity relations needed for adjoint-based inverse design by relating fields generated and measured at exchanged points. The derivation uses Maxwell’s equations, weak forms, and dipole-source symmetry.

  • Reciprocity question: The appendix asks whether fields at one point from a dipole at another can be recovered by exchanging the dipole and measurement points.This exchange question is represented in Figure A.1 and motivates the reciprocity derivations.
  • Role in inverse design: Rayleigh-Carson reciprocity relations provide crucial symmetry relations for the inverse-design procedure.The appendix gives an alternate derivation because these relations connect direct and adjoint field calculations.
  • Derivation framework: The derivation reduces Maxwell’s curl equations to second-order field equations and then to weak forms over a domain.Weak forms use test functions and integration by parts, with boundary conditions introduced for clarity while allowing more general conditions later.
  • Case structure: Four dipole-field combinations arise, but relations between mixed electric and magnetic cases reduce the number of unique cases to three.The appendix therefore treats electric-electric, magnetic-magnetic, and mixed electric-magnetic reciprocity separately.

Case I: The electric field from an electric dipole

The electric-electric reciprocity case shows that exchanging source and measurement points exchanges the corresponding field components. The result relies on the symmetry of the permittivity and permeability tensors, with transposed material properties needed more generally.

  • Setup: The electric-electric case places unit electric dipoles at two points and compares the resulting fields while keeping the background materials fixed.The weak-form equations are tested with the field generated by the other dipole.
  • Material assumptions: Electric-electric reciprocity requires symmetric permittivity and permeability tensors, although scalar materials automatically satisfy the condition.Even anisotropic, non-diagonal tensors are allowed when symmetric; otherwise the adjoint simulation uses transposed material properties.
  • Reciprocity result: The i-component at x2 from a j-oriented electric dipole at x1 equals the j-component at x1 from an i-oriented electric dipole at x2.This is the Green-function symmetry that permits exchanged dipole and measurement points.
  • Related case: The magnetic-magnetic case obeys an analogous exchanged-point relation under the same symmetric-material assumption.Its component-wise statement equates the magnetic field generated in the two exchanged configurations.

Case III: The electric field from a magnetic dipole (and vice versa)

The mixed electric-magnetic reciprocity case differs from the electric-electric case because exchanging source and measurement points also changes the field type and introduces a minus sign. The relation is derived using Maxwell’s equations, integration by parts, and symmetric material tensors.

  • Setup: The mixed case compares an electric dipole at x1 with a magnetic dipole at x2 rather than simply exchanging identical source and measurement types.The derivation requires careful treatment of the magnetic-dipole term and assumes no magnetic dipoles on the domain boundary.
  • Derivation: The derivation applies weak-form substitutions and Maxwell’s curl relation to obtain the final mixed reciprocity identity.Symmetry of the relevant material tensor is used in simplifying the resulting expression.
  • Reciprocity result: The magnetic field from an electric dipole equals the negative of the exchanged electric field from a magnetic dipole, component by component.Thus, unlike the electric-electric case, exchanging the source and measurement configuration changes both field type and sign.

Time-Dependent Merit Functions

The framework extends steady-state inverse design to time-dependent merit functions defined over both spatial and temporal domains. The resulting derivation uses time-dependent Green’s functions and preserves the shape-derivative approach.

  • Time-Dependent Objective: The resulting derivation extends the steady-state shape derivative to objectives that depend on time-dependent electromagnetic fields.The underlying geometry variation remains the source of the induced polarization used in the derivative.
  • Time-Dependent Objective: Time-dependent merit functions generalize steady-state objectives by evaluating arbitrary functions of fields over spatial domain χ and time domain T.The framework can target objectives such as maximizing absorption during T = [0, 100ns].
  • Shape-Derivative Derivation: The time-dependent derivation specifies merit-function variation with respect to field variation, then relates field changes to geometry-induced polarization through time-dependent Green’s functions.The induced polarization is integrated over space and time, with causality restricting contributing times to those before the observation time.
  • Shape-Derivative Derivation: Causality requires the induced polarization region T to contain only times before the observation time, generally giving T = (−∞, t).This temporal restriction determines which earlier fields contribute to the derivative.

APPENDIX B. TIME-DEPENDENT MERIT FUNCTIONS

For time-dependent objectives, the adjoint formulation is determined by a symmetry relation that exchanges present and past times. The resulting shape derivative is evaluated using an adjoint simulation run backward in time.

  • Adjoint Construction: The adjoint simulation must run backward in time, with sources driven by the time-dependent electric and magnetic fields.The backward evolution follows directly from the time-reversed Green’s-function symmetry.
  • Green’s-Function Symmetry: Time-dependent Green’s-function symmetry relates a source at time t to a measurement at an earlier time t′ < t.This reverses the temporal ordering used in the adjoint construction.
  • Adjoint Construction: The adjoint field at (x′, t′) is generated by sources derived from the merit-function derivatives with respect to the electric and magnetic fields.The source pair is specified as (P, M) = (∂f/∂E, −∂f/∂H).
  • Final Derivative: The time-dependent shape/topological derivative is obtained by integrating induced polarization dotted with the adjoint field over space and time.This expression is the time-dependent counterpart of the steady-state derivative.
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