Nanophotonic light trapping in solar cells

TL;DR: In this article, the authors review the theory of nanophotonic light trapping, with experimental examples given where possible, focusing particularly on periodic structures, since this is where physical understanding is most developed, and where theory and experiment can be most directly compared.

Abstract: Nanophotonic light trapping for solar cells is an exciting field that has seen exponential growth in the last few years. There has been a growing appreciation for solar energy as a major solution to the world’s energy problems, and the need to reduce materials costs by the use of thinner solar cells. At the same time, we have the newly developed ability to fabricate controlled structures on the nanoscale quickly and cheaply, and the computational power to optimize the structures and extract physical insights. In this paper, we review the theory of nanophotonic light trapping, with experimental examples given where possible. We focus particularly on periodic structures, since this is where physical understanding is most developed, and where theory and experiment can be most directly compared. We also provide a discussion on the parasitic losses and electrical effects that need to be considered when designing nanophotonic solar cells.

Summary

VII. PERIODIC LIGHT TRAPPING

• Given that earth receives more energy from Sun in one day (1021 J) than is used by the world population in one year, PV contribution to the world energy has vast potential.
• C-Si solar cells have now exceeded efficiency of 25% in the laboratory, and silicon modules have reached an efficiency of over 22%.3 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions the cost per watt of PV generated electricity would be to design and fabricate high efficiency solar cells based on thin active layers.
• Light trapping is achieved by modifying the surface of the solar cell to enhance the probability of total internal reflection.
• Periodic structures can provide high enhancements in their own right.

II. LAMBERTIAN LIGHT TRAPPING

• Among the earliest work on the theory of light trapping was the derivation by Yablonovitch and Cody,10 and independently by G€oetzberger11 of the light trapping achievable by an isotropically scattering (or Lambertian) surface.
• The same result can be obtained using a statistical approach.
• Another important assumption is that the optical mode density in the structure is continuous and is unaffected by wave-optical effects.
• Thus, the average path length for a single pass across the semiconductor for rays scattered by a Lambertian surface is ðp2 0 w cos h cos h sin hdh ðp2 0 cos h sin hdh ¼ 2w: (2) The above two results can be used to calculate the average path length enhancement resulting from any Lambertian scheme.

III. EFFECT OF LIGHT TRAPPING ON REQUIRED THICKNESS FOR STRONG AND WEAK ABSORBERS

• The level of light trapping that can be achieved determines the thickness that is needed to achieve an adequate level of Jsc. Figure 4 shows the Jsc as a function of thickness for Si and GaAs, with either no light trapping (single pass absorption) or Lambertian light trapping.
• Note that the use of single pass absorption and Lambertian light trapping is not strictly valid for thicknesses below a few hundred nanometres, because waveguide effects need to be taken into account.
• Nevertheless, this simple calculation allows us to estimate the thickness required for a given semiconductor and level of light trapping.
• The ability of light trapping to reduce the required thickness of direct bandgap semiconductors to tens of nanometres opens up the possibility of using a wide range of alternative materials for photovoltaics that have very low diffusion lengths, such as CuO and FeS2.

IV. EFFECT OF LOSSES

• When considering the potential Jsc or path length enhancement in a solar cell, it is very important to also consider the effect of parasitic loss within the cell, due to, for example, absorption within a rear reflector.
• Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions enhancement, for different values of thickness and Rf.
• The authors can see that if the loss at the rear reflector is very low (1%), the path length enhancement for the Lambertian case is close to the ideal value of 4n2 50.
• The authors can see that for a 100 lm thick silicon solar cell, there is no significant benefit in above Lambertian light trapping, regardless of the loss at the rear reflector.
• This would seem to preclude the use of metals and heavily doped semiconductors in the cell design, unless the amounts used are very small.

V. ELECTRICAL EFFECTS

• The inclusion of light trapping reduces the optimum thickness of cells, as this allows high currents to be maintained but with lower bulk recombination.
• The lesson here is that for ultra-thin device concepts, an estimate of the effect of surface recombination needs to be made before doing detailed optical modeling.
• The effect of photon emission efficiency, gext, on open circuit voltage can be seen from Eq. (7), derived by Ross,18 and discussed further in Refs. 16, 17, and 19: Voc ¼ Voc ideal ðkT=qÞlnðgextÞ: (7) For a thin cell in the geometric optics limit and no losses, gext¼.
• In practice, the effect has been lower to date.
• For practical applications, the electrical performance of the solar cell in the presence of a grating or photonic crystal structure also needs be considered.

VI. MEASURING LIGHT TRAPPING

• The most common method to evaluate light trapping for nanophotonic structures is to calculate the enhancement factor, i.e., the photocurrent at long wavelengths divided by the photocurrent for a reference sample.
• An alternative method is to measure the Jsc for the test sample and compare it to the Jsc for a reference sample.
• Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions trapping, since the enhancement in Jsc that is achievable depends strongly on the solar cell thickness.
• This involves plotting the inverse IQE against the inverse absorption co-efficient.
• Trupke et al. have presented a method for obtaining the useful absorptance and the light trapping enhancement factor Z from measurements of luminesence.

VII. PERIODIC LIGHT TRAPPING STRUCTURES

• One, two, or three dimensional (1D, 2D, or 3D) periodic dielectric structures or gratings have the potential to enhance the optical absorption of solar cells in several ways.
• Optimised 1D dielectric gratings or Bragg stacks can be used as back reflectors that double the path length of light in the active volume of a solar cell ).
• Light trapping can be achieved by either coupling light into the guided modes of the active region (if the active region supports only a few waveguided modes) or by coupling light into diffraction modes that propagate outside the loss cone in the active volume (when the active region is relatively thick and supports a continuous density of photonic modes).
• Alternatively, the active volume itself can be patterned in 3D to confine light and increase the absorption of long wavelength light28–33 ).
• Several gratings can be used in conjunction to each other to achieve one or more of the above mentioned effects simultaneously.

A. Gratings for back-reflectors and anti-reflection

• Light reflected from different layers of the stack interfere constructively or destructively leading to high reflectance or transmittance.
• Gratings can provide anti-reflection by two mechanisms.
• For gratings with periodicity d k, incident plane waves can couple to only one propagating diffraction mode in air, the principal diffraction order or the 0th diffraction order.
• Each grating mode is characterized by an effective refractive index, neff.
• Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions modes for 2D sub-wavelength scale gratings.53 Detailed analysis can be found in Ref. 54.

B. Gratings for light trapping

• Two-dimensional (2D) gratings with periods larger than that of the incident wavelength (in the medium) support higher order diffraction modes that propagate at an angle with respect to the surface normal.
• Light trapping can be achieved by designing the gratings such that most of the light incident on the gratings is coupled to diffraction orders propagating outside the escape cone in the active volume of the solar cell.
• The red and blue circles represent the range of k-vectors for propagating modes inside the absorber layer for light of wavelength k1 and k2, respectively, such that k1> k2.
• For large period gratings ), the relative change in the number of propagating diffraction modes is very small compared to the relative change in the number of propagating diffraction modes for small period gratings ) when the wavelength of incident light is varied.
• Haase and Stieberg77 have predicted through simulations that the short circuit current density of a 1 lm thick c-Si solar cell can be increased by 18.7% using rectangular gratings and by 28.2% using 6 step blazed gratings.

C. Understanding rectangular and pillar gratings

• Rectangular gratings and pillar gratings are simple structures that are technologically important because they can be fabricated relatively easily.
• It has been shown that simplified modal analysis52,81,82 can be used to predict the optimal parameters for rectangular gratings for coupling most of the incident light into diffraction orders propagating outside the loss cone.
• It can be seen that there are periodic peaks in the diffuse transmittance, for a given wavelength.
• Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions h ¼ k 2nef f0 : (12) When the period of the grating is such that at least two grating modes are excited (for normal incidence, this corresponds to 0th and 3rd grating modes), interference between the grating modes determines the efficiency of excitation of various diffraction modes.
• Light is efficiently coupled into the 0th diffraction order or the principal diffraction order when the grating height, h is such that the grating modes accumulate a net phase difference of zero.

D. Design of large period gratings

• Modal analysis is very effective in predicting the behaviour of rectangular gratings or pillar gratings.
• So, the transmitted far-field light distribution due to the grating can now be evaluated from the Fourier transforms of the individual transmission functions represented in Eqs. (13) and (14).
• Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions.
• When the grating height is increased to 150 nm, the transmission peak from the single prism is centered between the 0th and 1 diffraction orders of the periodic structure, resulting in equal intensity distribution among the 0 and 1 diffraction orders for the grating.
• For a grating height of 500 nm, a large fraction of the transmitted light is trapped inside Si.

E. Fundamental limits with gratings

• The above approaches are simple means of optimizing a large period (d> k) grating performance.
• Moreover, the wavevector of the propagating modes should lie within a circle of radius |k0|, the wavevector of incident light ).
• This upper bound on the path length enhancement is lower than the Lambertian limit of 4n2, but equal to the 2D enhancement limit for the case of geometrical optics.
• The minimum number of free space propagating modes for a bi-period grating are N¼ 2, one for each polarisation.
• Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions total number of guided optical modes supported by the absorber in the frequency range (x, xþDx) is M ¼ 8n 3px2 c3 d 2p 2 t 2p Dx: (21) Using Eqs. (18) and (21), the upper bound on the maximum path length enhancement achievable with bi-periodic gratings is 4pn2.80.

F. Angular dependence and light trapping

• Using geometrical optics, it can be shown that the path length enhancement increases from 4n2 to 4n2/sin2h for a restricted acceptance half-angle of h.91.
• The relation 4n2/sin2h comes fundamentally from the conservation of etendue and radiance, valid for layers that are sufficiently thick that they support many optical modes.
• Their analysis also shows that the angle-integrated upper limit for the absorption enhancement is lower for small period gratings compared to large period structures.
• The position of the diffraction modes for off-normal incidence is shifted with respect to that for normal incidence and is represented by open dots.
• So, small period gratings can effectively couple only to normally incident light.

G. Asymmetric gratings

• Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions structures can approach that limit.
• The theoretical approaches presented in Sec. VII F can be used to determine whether asymmetry in gratings is beneficial for light trapping.
• These results are consistent with the extensive analysis presented by Yu et al. for estimating the upper bound of absorption enhancement using gratings.
• For a symmetric grating, the guided optical modes in the absorber either have an odd or even modal profile.

VIII. OTHER LIGHT TRAPPING STRUCTURES

• There are a variety of other structures that can be used for nanophotonic light trapping in solar cells.
• The most common methods are plasmonic structures, random scattering surfaces, and nanowires.
• In the case of plasmonics, this is because there have already been several good reviews published recently.
• Nanowires are a younger subject, where very impressive optical enhancements have been achieved.
• The major challenge in this field is likely to be electrical rather than optical.

A. Plasmonic structures

• Plasmonic light trapping involves either localized surface plasmons supported by discrete metal nanoparticles or surface plasmon polaritons supported by continuous metal films.
• Discrete metal nanoparticles can be used as efficient scatterers to couple incident sunlight into trapped modes in thin solar cells.
• This was first demonstrated by Stuart and Hall101 and subsequently by several other researchers.
• A detailed overview of plasmonic light trapping can be found in Refs. 115–118.

B. Random scattering surfaces

• Random scattering surfaces are easy to fabricate and effectively trap light over a wide angular and wavelength range with minimal parasitic absorption.
• It has been shown that it will be advantageous to direct light outside the escape cone124 and that criteria for the depth of local features can be derived.
• As with grating structures, increased surface area can lead to increased surface recombination and steep features can lead to reduced material quality if deposited on a textured surface.

D. Fundamental limits with general nanophotonic structures

• In the above sections, the authors first described the fundamental limit for bulk semiconductors with isotropically scattering, geometrical scale features (the Lambertian case of 4n2 enhancement).
• The optical mode density in air or in the absorbing layer is not altered.
• Hence, for both types of structures, there is a trade-off between enhancement at normal incidence and the angular response.
• Callahan et al. have shown that light trapping beyond the 4n2 limit can be achieved provided that the structure has an increased local density of optical states (LDOS) compared to the bulk semiconductor.

IX. SUMMARY

• Nanophotonics is essential for increasing the absorption in thin film solar cells.
• Periodic photonic structures are important in their own right and also provide the best developed starting point for insight into the physical mechanisms involved in nanophotonic light trapping.
• Other structures are also very promising for light trapping, and some of the ideas developed for periodic structures are also transferrable to these contexts.
• This field is developing very rapidly, and there may also be interesting synergies with other areas of photovoltaics that have not yet been imagined.

Figures

FIG. 12. Contour plot of diffuse transmittance vs wavelength and height for a rectangular Si grating on an infinite Si substrate with L¼ 650 nm calculated with RCWA. The lines show the maxima predicted with modal analysis. Reprinted with permission from K. R. Catchpole, J. Appl. Phys. 102(1), 013102 (2007). Copyright # 2007 American Institute of Physics.

FIG. 16. (a) and (b) Schematic demonstration of the effect of grating period on the angular dependence of its response. The solid green dots represent the allowed diffraction modes in k-space (reciprocal space) for normal incidence. The hollow green dots represent the shift in the position of the modes for offnormal incidence. The areas bounded by the circles represent the allowed inplane k vectors for the propagating modes supported by the medium. The radii of the circles are jkj.

FIG. 6. Schematic illustration of periodic dielectric structures used for enhancing solar cell performance: (a) 1D (Bragg) stacks used as back reflectors, (b) 2D gratings, used for anti-reflection and/or light trapping (c), and (d) 3D photonic crystals, which are highly effective as omnidirectional reflectors or trap light by a waveguiding effect. Reproduced with permission from Catchpole et al., MRS Bull. 36, 461 (2009). Copyright # 2011 Cambridge University Press.

FIG. 11. Effect of incorporating wavelength scale diffraction gratings on the short circuit current density of a crystalline silicon solar cell. Ideal (theoretically evaluated) current densities expected from a solar cell for single pass, with an ideal rear reflector and with a Lambertian light trapping are also shown for reference. A perfect anti-reflection layer is assumed for these cases. The symbols in red correspond to the reference solar cell and the same symbol in green corresponds to the case where gratings are used to enhance the solar cell performance. Open symbols correspond to numerical results (up-triangles,64 down triangles,64 diamonds and crossed diamond,68 and squares69) and filled symbols correspond to experimental results (circles,62 stars,66 triangles,74 squares,71 and diamonds72).

FIG. 10. (a) and (b) Schematic demonstration of the effect of grating period, d on the wavelength dependence of the grating performance. The dots represent the allowed diffraction modes in k-space (reciprocal space) for normal incidence. The areas bounded by the circles represent the allowed in-plane k vectors for the propagating modes supported by the medium. The radii of the circles are jk1j or jk2j.

FIG. 1. The spectral irradiance on earth’s surface, irradiance absorbed by a 2 lm thick Si layer and a 2 lm thick Si with Lambertian light trapping, neglecting the reflection losses.

FIG. 2. A glass slide with an imprinted light trapping pattern on the surface. The pattern traps light effectively inside the glass, making it bounce back into the glass everytime it hits the glass-air interface. As a result, light is trapped and is directed to the edges of the glass slide, as can be seen here.

TABLE I. Summary of maximum limit for light trapping and analysis techniques for different structures.

FIG. 14. Normalized transmitted intensity distribution of a Si grating with a period, d 1500 nm. The intensity distribution of the periodic arrangement (obtained using Eq. (16)) is represented in red, while the intensity distribution from a single prism (obtained using Eq. (17)) is represented in blue and the net intensity (evaluated using Eq. (15)) is represented in black. The wavelength of the incident light is fixed at 800 nm and the grating height is (a) 50 nm, (b) 150 nm, (c) 300 nm, (d) 500 nm, or (e) 600 nm. Reproduced with permission from S. Mokkapati et al., J. Phys. D: Appl. Phys. 44(5), 055103 (2011). Copyright # 2011 Institute of Physics.

FIG. 8. Schematic showing the parameters of (a) 1D lamellar grating and (b) 2D pillar grating used for illustrating the methods of determining the grating modes (the photonic crystal equation for (a) and the effective index method for (b)).

FIG. 7. (a) Schematic of a 2D grating structure with periodicity d. (b) Allowed diffraction modes in k-space or reciprocal space. Circles with radii r1 and r2 represent the maximum allowed k vectors for propagating modes in air and the absorbing medium, respectively. n is the refractive index of the absorbing medium.

FIG. 3. Definition of angles associated with light incident on a perfectly randomizing surface. Adapted from Ref. 8.

FIG. 13. The grating structure considered for demonstration of the analytical approach. The transmission function of the grating is represented as the convolution of the transmission function of a periodic structure with periodicity d, and a triangular component with base d and height h. N is the number of periods considered for calculations and n is the refractive index of the grating material. The expressions for the transmission functions are indicated below the corresponding structure. Reproduced with permission from S. Mokkapati et al., J. Phys. D: Appl. Phys. 44(5), 055103 (2011). Copyright # 2011 Institute of Physics.

FIG. 9. Moth eye structure fabricated using electron beam lithography and reactive ion etching for application as an anti-reflection layer. Reproduced with permission from Sai et al., Prog. Photovoltaics 15(5), 415 (2007). Copyright # 2007 John Wiley & Sons Ltd.

FIG. 4. Jsc vs. thickness for a direct bandgap semiconductor (GaAs) and an indirect bandgap semiconductor (Si), for either the planar case (assuming single pass absorption only) or Lambertian light trapping.

FIG. 5. (a) Path length enhancement vs. loss at the rear reflector (1 Rb). The blue solid line is for Lambertian light trapping (Rf¼ 1 1/n2), where n is the refractive index of silicon. The red dashed line is for an internal front reflectivity loss with 10% of the value for Lambertian light trapping, i.e., Rf¼ 1 0.1/n2. (b) Jsc vs. loss at the rear reflector (1 Rb). The blue solid lines are for Lambertian light trapping, while the red dashed lines are for above Lambertian light trapping, as in (a). Three thicknesses of silicon are shown: 100 lm, 10 lm, and 1 lm.

FIG. 17. Schematic illustration of (a) out-coupling of light from a solar cell with a symmetric grating structure and (b) reduced out-coupling by introducing asymmetry into the grating structure.

Full text

Nanophotonic light trapping in solar cells
S. Mokkapati and K. R. Catchpole
Citation: J. Appl. Phys. 112, 101101 (2012); doi: 10.1063/1.4747795
View online: http://dx.doi.org/10.1063/1.4747795
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v112/i10
Published by the American Institute of Physics.
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APPLIED PHYSICS REVIEWS—FOCUSED REVIEW
Nanophotonic light trapping in solar cells
S. Mokkapati
1
and K. R. Catchpole
2
1
Department of Electronics Materials Engineering, Research School of Physics and Engineering,
Australian National University, Canberra ACT-0200, Australia
2
Center for Sustainable Energy Systems, Australain National University, Canberra ACT-0200,
Australia
(Received 17 April 2012; accepted 8 August 2012; published online 19 November 2012)
Nanophotonic light trapping for solar cells is an exciting field that has seen exponential growth in
the last few years. There has been a growing appreciation for solar energy as a major solution to
the world’s energy problems, and the need to reduce materials costs by the use of thinner solar
cells. At the same time, we have the newly developed ability to fabricate controlled structures on
the nanoscale quickly and cheaply, and the computational power to optimize the structures and
extract physical insights. In this paper, we review the theory of nanophotonic light trapping, with
experimental examples given where possible. We focus particularly on periodic structures, since
this is where physical understanding is most developed, and where theory and experiment can be
most directly compared. We also provide a discussion on the parasitic losses and electrical effects
that need to be considered when designing nanophotonic solar cells.
V
C
2012 American Institute of
Physics.[http://dx.doi.org/10.1063/1.4747795]
TABLE OF CONTENTS
I. INTRODUCTION ............................ 1
II. LAMBERTIAN LIGHT TRAPPING . . . ........ 3
III. EFFECT OF LIGHT TRAPPING ON
REQUIRED THICKNESS FOR STRONG AND
WEAK ABSORBERS . . ..................... 4
IV. EFFECT OF LOSSES . . ..................... 4
V. ELECTRICAL EFFECTS ..................... 5
VI. MEASURING LIGHT TRAPPING . . . . ........ 5
VII. PERIODIC LIGHT TRAPPING
STRUCTURES ............................ 6
A. Gratings for back-reflectors and anti-
reflection ............................. 6
B. Gratings for light trapping .............. 8
C. Understanding rectangular and pillar
gratings............................... 10
D. Design of large period gratings . . ........ 11
E. Fundamental limits with gratings . ........ 13
F. Angular dependence and light trapping . . . . 14
G. Asymmetric gratings ................... 14
VIII. OTHER LIGHT TRAPPING STRUCTURES . . 15
A. Plasmonic structures ................... 15
B. Random scattering surfaces . . . . . ........ 16
C. Nanowires............................ 16
D. Fundamental limits with general
nanophotonic structures . . .............. 16
IX. SUMMARY................................ 17
I. INTRODUCTION
Worldwide installed capacity of solar photovoltaic (PV)
power has escalated from 1.3 GW in 2001 to 22.9 GW in
2009 and 35 GW in 2010 (Ref. 1) and has had an average
growth rate of 40% over the last 5 years. Given that earth
receives more energy from Sun in one day (10
21
J) than is
used by the world population in one year, PV contribution to
the world energy has vast potential. Currentl y, the cost per
Watt of power generated from PV is substantially higher
than the current costs of power generated using conventional
fossil fuels. Cost of power generation from PV needs to be
reduced approximately by a factor of 2-3 (exact values
depend on local solar insolation and electricity costs) to be
comparable to that generated from conventional s ources.
Currently, 80%–90% of the PV market is based on crys-
talline Si (c-Si) solar cells.
2
Si is the third most abundant ele-
ment on Earth and has a near ideal band gap energy for
maximizing the efficiency of a single junction solar cell. Si
fabrication techniques developed by the microelectronics
industry have played a critical role in the development of
current c-Si solar cell technology. c-Si solar cells have now
exceeded efficiency of 25% in the laboratory, and silicon
modules have reached an efficiency of over 22%.
3
High purity Si used in the fabrication of conventional
c-Si solar cells requires expensive and energy intensive refin-
ing of the Si feedstock. Materials costs account for 40% of
the total cost of a typical c-Si PV module. Only 25% of the
total costs are spent on actual cell fabrication. The rest is the
module fabrication cost. An effective approach to reducing
0021-8979/2012/112(10)/101101/19/$30.00
V
C
2012 American Institute of Physics112, 101101-1
JOURNAL OF APPLIED PHYSICS 112, 101101 (2012)
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the cost per watt of PV generated electricity would be to
design and fabricate high efficiency solar cells based on thin
active layers. Conventional c-Si solar cells are fabricated
from 180–300 lm thick Si wafers. Fabricating thin film solar
cells with an active layer thickness of hundreds of nano-
meters to few microns would reduce the material usage by a
factor of 100. In addition to reduced materials usage, thin
film solar cells also have the advantage of reduced carrier
collection lengths. The photo-generated carriers in the cell
should reach the external contacts before they recombine in
order to generate electric current. The distance travelled by
the carriers before recombination is called carrier diffusion
length. For efficient collection of photo-generated carriers,
the carrier diffusion length in the active material should be a
few times larger than the thickness of the active layer.
Reduced thicknesses facilitate the use of lower quality active
material (material with lower carrier diffusion lengths) for
the cell fabrication, further reducing the material and deposi-
tion costs.
Currently, CdTe (cadmium Telluride) holds the major
share in thin film PV market. However, Tellurium is a rare
material and might not be a suitable candidate to provide a
significant fraction of the world’s projected energy needs in
the future. Si based thin-film technologies like a-Si:H
(hydrogenated amorphous Si), polycrystalline Si and tandem
microcrystalline Si, a-Si:H solar cell technologies are cur-
rently under intense investigation. Other potential candidate
materials are organics and earth-abundant inorganic semi-
conductors such as copper-zinc-tin-sulphur (CZTS), CuO,
and FeS
2
.
A thin active layer, however, compromises the optical
absorption in the solar cell. Figure 1 shows the spectral irra-
diance (in W/m
2
/nm) on earth’s surface for the AM1.5g solar
spectrum, and the irradiance absorbed by a 2 lm thick Si
layer, neglecting reflection losses (i.e., assuming a perfect
anti-reflection coating on the front surface). As can be seen
from Figure 1, for wavelengths greater than 500 nm, not all
of the incident photons are absorbed in the Si layer. Part of
the incident energy is lost because of transmission of light
through the Si layer. These transmission losses are more sig-
nificant in the long wavelength region, 700 nm–1180 nm,
closer to the band-edge of Si. The transmission losses can be
reduced by “folding” light multiple times into the absorbing
region of the solar cell, thereby increasing the optical path
length of light and hence the probability of its absorption
inside the solar cell. This process is known as light trapping.
By employing light trapping in a solar cell, the “optical
thickness” of the active layer is increased several times while
keeping its physical thickness unaltered. The ratio of the op-
tical thickness to the physical thickness, i.e., the ratio of the
path length travell ed by photons inside the cell in the pres-
ence of light trapping to that with no light trapping, is known
as the path length enhancement. This is an important param-
eter that enables quantitative comparison of different light
trapping techniques.
Light trapping is achieved by modifying the surface of
the solar cell to enhance the pr obability of total internal
reflection. By doing so, light gets reflected back into the
active volume several times. Figure 2 shows a glass slide
with a light trapping pattern imprinted on its surface. The
outline of the pattern can be seen clearly on the surface of
the slide. When light hits the glass-air interface, it is reflected
back completely into glass and cannot escape through the
surface. Light can only escape the glass slide from its edges,
making the edges colorful, as can be seen in the picture.
The best known technique for texturing the surface of a
Si wafer is to form upright or inverted pyramids
4–8
or ran-
dom textures.
9
This is the most widely used light trapping
technique employed in industry. These pyramids have
dimensions of the order of 1–10 lm. However, this approach
is not suitable for thin solar cells where the active region
itself is only a few microns or even a few hundred nano-
meters thick.
Thin solar cells need wavelength scale (or nanopho-
tonic) structures for achieving light trapping. Different
approaches for nanophotonic light trapping include periodic
semiconductor or dielectric structures (one dimensional
(1D), single period or dual period two dimensional (2D)
diffraction gratings and photonic crystals), plasmonic
FIG. 1. The spectral irradiance on earth’s surface, irradiance absorbed by a
2 lm thick Si layer and a 2 lm thick Si with Lambertian light trapping,
neglecting the reflection losses.
FIG. 2. A glass slide with an imprinted light trapping pattern on the surface.
The pattern traps light effectively inside the glass, making it bounce back
into the glass everytime it hits the glass-air interface. As a result, light is
trapped and is directed to the edges of the glass slide, as can be seen here.
101101-2 S. Mokkapati and K. R. Catchpole J. Appl. Phys. 112, 101101 (2012)
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structures, and randomly structured semiconductor surfaces.
The tremendous research efforts in the direction of designing
best possible light trapping strategies for photovoltaics is
evident in the fact that a search for publications in Scopus in
2001 for solar and light-trapping yields 24 publications,
while the same search for 2011 yields 242 publications.
In this article, we provide a general introduction to
nanophotonic light trapping in solar cells. We describe the
potential for enhancement that they provide, including the
work that has been done to date on the fundamental limits of
these structures. We also describe the effect of losses and the
measurement of light trapping, which are areas that have not
received much attention to date. We give particular attention
to periodic structures, for two important reasons. Periodic
structures can provide high enhancements in their own right.
In addition, there is very well developed theory for structures
with very small or very large periods compared to the wave-
length of incident light. Recent advances in numerical
techniques for wav elength scale gratings help in gaining fun-
damental understanding of the mechanisms involved.
Advances in processing technologies in recent times also
make it possible to fabricate sub-wavelength scale structures
with precisely controllable features and allow a direct com-
parison between theory and experiment for periodic struc-
tures. We also provide a brief discussion of the light trapping
provided by other structures, and point to where more infor-
mation on these topics can be found.
II. LAMBERTIAN LIGHT TRAPPING
Among the earliest work on the theory of light trapping
was the derivation by Yablonovitch and Cody,
10
and inde-
pendently by G
€
oetzberger
11
of the light trapping achievable
by an isotropically scattering (or Lambertian) surface. A
Lambertian surface results in uniform brightness in a me-
dium, irrespective of the angle of incidence of light on the
medium, or the angle of observation. In this section, we will
derive a simple expression for the path length enhancement
that can be obtained in a solar cell using a Lambertian
surface.
We follow the approach outlined in Refs. 8 and 11. The
same result can be obtained using a statistical approach.
12
The approach used here is valid only in weakly absorbing
limit, i.e., there is negligible absorption inside the structure
or on the surface of the structure. Another important assump-
tion is that the optical mode density in the structure is contin-
uous and is unaffected by wave-optical effects. These
conditions are satisfied if the optical thickness of the cell is
much greater than k/2 n, where k is the wavelength of inci-
dent light and n is the refractive index of the material; and
the surface texture is either random or has a period much
larger than the waveleng th, k, of incident light.
It is assumed that the Lambertian surface has produced a
uniform brightness (intensity per solid angle) of B inside the
semiconductor. Any unit surface area of the semiconductor
will have incident intensity B cos h per unit solid angle from
rays oriented at an angle h to its normal (see Figure 3). The
proportion of light intensity incident on the surface that
escapes from the surface, f, is the ratio of intensity resulting
from light rays within the loss cone (i.e., rays striking the
surface at an angle smaller than the critical angle, h
c
) to the
total intensity.
f ¼
ð
h
c
0
B cos h sin hdh
ð
p
2
0
B cos h sin hdh
¼ sin
2
h
c
¼
1
n
2
; (1)
where n is the refractive index of the semiconductor.
In the absence of the Lambertian surface, normally inci-
dent light on a semiconductor substrate of thickness w would
travel an average distance of w in one pass across the semi-
conductor. Because of the Lambertian scattering surface,
light is scattered into the semiconductor at all angles with
respect to the surface normal. A ray traversing at an angle h
with respect to the surface normal propagates a distance
w/cosh in one pass across the semiconductor. Thus, the aver-
age path length for a single pass across the semiconductor
for rays scattered by a Lambertian surface is
ð
p
2
0
w
cos h
cos h sin hdh
ð
p
2
0
cos h sin hdh
¼ 2w: (2)
The above two results can be used to calculate the aver-
age path length enhancement resulting from any Lambertian
scheme. For a solar cell with front Lambertian surface and a
rear reflector with reflectivity, R, the average path length is
given by
2wð1  RÞþ4wfR þ 6 wR ð1  f Þð1  RÞ
þ 8wf ð1  f ÞR
2
þ :::::::: ¼
2wð1 þ RÞ
1  Rð1  f Þ
: (3)
For the limiting case of a perfect rear reflector, i.e.,
R ¼ 1, the average path length becomes 4n
2
w; or the average
path length enhancement with respect to a planar semicon-
ductor of thickness w is 4n
2
. The same argument can be
FIG. 3. Definition of angles associated with light incident on a perfectly ran-
domizing surface. Adapted from Ref. 8.
101101-3 S. Mokkapati and K. R. Catchpole J. Appl. Phys. 112, 101101 (2012)
Downloaded 25 Mar 2013 to 150.203.178.118. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

followed for isotropic scattering in 2D rather than 3D, i.e., a
plane normal to the scattering surface. For this case, by omit-
ing sin h in the integrations in Eqs. (1) and (2) above, a path
length enhancement of pn is obtained. A good overview of
Lambertian and geometrical optics light trapping can be
found in Brendel,
13
who also shows that for maximum cur-
rent, the distribution of path lengths as well as the average
path length is important.
As already mentioned, the above path length enhance-
ment estimations are valid only for thick substrates with sur-
face structures that can be described by geometrical optics.
We will see in Secs. VII E and VIII D how wave-optics affect
these results.
III. EFFECT OF LIGHT TRAPPING ON REQUIRED
THICKNESS FOR STRONG AND WEAK ABSORBERS
The level of light trapping that can be achieved deter-
mines the thickness that is needed to achieve an adequate
level of J
sc
. Figure 4 shows the J
sc
as a function of thickness
for Si and GaAs, with either no light trapping (single pass
absorption) or Lambertian light trapping. Note that the use of
single pass absorption and Lambert ian light trapping is not
strictly valid for thicknesses below a few hundred nano-
metres, because waveguide effects need to be taken into
account. Nevertheless, this simple calculation allows us to
estimate the thickness required for a given semiconductor
and level of light trapping. We can see from Figure 4 that
for an indirect bandgap semiconductor like Si, light
trapping decreases the thickness required to achieve a J
sc
of
35 mA/cm
2
from about 40 lm to about 2 lm. For a direct
bandgap semiconductor such as GaAs, the thickness required
to reach 28 mA/cm
2
is reduced from 1 lmto50nm.The
ability of light trapping to reduce the required thickness of
direct bandgap semiconductors to tens of nanometres opens
up the possibility of using a wide range of alternative materi-
als for photovoltaics that have very low diffusion lengths,
such as CuO and FeS
2
.
IV. EFFECT OF LOSSES
When considering the potential J
sc
or path length
enhancement in a solar cell, it is very important to also con-
sider the effect of parasitic loss within the cell, due to, for
example, absorption within a rear reflector.
To investigate this, we can use the analytical model for
Lambertian light trapping given by Green,
14
valid for rela-
tively thick cells that support a large number of waveguide
modes. The total absorption in the cell, A
T
, the absorption in
the rear reflector, A
R
and the absorption in the bulk region of
a cell, A
B
are then given by
A
T
¼
ð1  R
ext
Þð1  R
b
T
þ
T

Þ
1  R
f
R
b
T
þ
T

; (4)
A
R
¼
ð1  R
ext
Þð1  R
b
T
þ
Þ
1  R
f
R
b
T
þ
T

; (5)
A
B
¼ A
T
 A
R
; (6)
where R
ext
is the external front reflectance, R
b
is the internal
rear reflectance, and R
f
is the internal front reflectance. T
þ
(T

) is the fraction of downward (upward) light transmitted
to the rear (front) surface, and expressions for these are given
in Ref. 14.
In Figure 5, we set R
ext
¼ 0 and investigate the effect of
loss at the rear reflector (1  R
b
) on the J
sc
and path length
FIG. 4. J
sc
vs. thickness for a direct bandgap semiconductor (GaAs) and an
indirect bandgap semiconductor (Si), for either the planar case (assuming
single pass absorption only) or Lambertian light trapping.
FIG. 5. (a) Path length enhancement vs. loss at the rear reflector (1  R
b
).
The blue solid line is for Lambertian light trapping (R
f
¼ 1  1/n
2
), where n
is the refractive index of silicon. The red dashed line is for an internal front
reflectivity loss with 10% of the value for Lambertian light trapping, i.e.,
R
f
¼ 1  0.1/n
2
. (b) J
sc
vs. loss at the rear reflector (1  R
b
). The blue solid
lines are for Lambertian light trapping, while the red dashed lines are for
above Lambertian light trapping, as in (a). Three thicknesses of silicon are
shown: 100 lm, 10 lm, and 1 lm.
101101-4 S. Mokkapati and K. R. Catchpole J. Appl. Phys. 112, 101101 (2012)
Downloaded 25 Mar 2013 to 150.203.178.118. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Frequently asked questions

Q1. What have the authors contributed in "Nanophotonic light trapping in solar cells" ?

In this paper, the Simicrograting solar cell with vertical sidewall electrodes was used to improve photovoltaic performance. 

Q2. What is the upper bound of the maximum path length enhancement achieved by gratings?

For single period, wavelength scale gratings, the number of free space propagating modes is still 1, but the number of accessible guided modes is halved due to symmetry constraints, reducing the upper bound on the maximum path length enhancement achievable to pn. 

Q3. What is the effect of introducing asymmetry into the grating structure?

For large period gratings (d> k), higher order diffraction modes exist in air and introducing asymmetry into the grating structure does not reduce out-coupling of light. 

Q4. What is the key advantage of large period diffraction gratings?

Key advantages of relatively large period diffraction gratings is that they are less wavelength sensitive, which is important in achieving light trapping for thin films which are weakly absorbing over a broad wavelength range. 

Q5. What is the maximum path length enhancement for the Lambertian case?

A reduced mode density in the active layer means reduced intensity and hence the maximum path length enhancement that can be achieved in this case is smaller than 4n2.142Again, assuming that the efficiency of coupling light into all available optical modes remains the same, the path length enhancement for the Lambertian case can be exceeded by either reducing the number of optical modes in air or by increasing the number of optical modes outside the escape cone in the absorbing layer or both. 

Q6. What are the main advantages of using subwavelength scale periodic dielectric structures?

Subwavelength scale periodic dielectric structures for anti-reflection layers have been fabricated using techniques like electron beam lithography followed by reactive ion etching,36 fast atom beam etching through alumina templates.60 Cheap and large area fabrication techniques like embossing ina sol-gel,44,49 or using spin coated or self-assembled dielectric spheres as etch patterns/masks38,39,48 have also been developed for fabrication of subwavelength scale gratings that could be employed as anti-reflection layers on solar cells. 

Q7. What is the effect of light trapping on the thickness of direct bandgap semiconductors?

The ability of light trapping to reduce the required thickness of direct bandgap semiconductors to tens of nanometres opens up the possibility of using a wide range of alternative materials for photovoltaics that have very low diffusion lengths, such as CuO and FeS2. 

Q8. How many path length enhancements have been demonstrated for silicon nanowire arrays?

131 Path length enhancements of up to 73 have been demonstrated for normalincidence for silicon nanowire arrays on silicon substrates. 

Q9. How can the authors predict the optimal parameters for rectangular gratings?

It has been shown that simplified modal analysis52,81,82 can be used to predict the optimal parameters for rectangular gratings for coupling most of the incident light into diffraction orders propagating outside the loss cone. 

Q10. What is the fundamental limit to light trapping given by a relatively thick waveguide?

Schiff has also shown that for a relatively thick waveguide on a metal, the fundamental limit to light trapping is given by 4n2þ nk/h, with the extra enhancement due to a plasmonic mode at the metal semiconductor interface. 

Q11. How much is the path length enhancement for the Lambertian case?

The authors can see that if the loss at the rear reflector is very low (1%), the path length enhancement for the Lambertian case is close to the ideal value of 4n2 50. 

Q12. How can absorber layers be designed to support enhanced density of optical modes?

absorber layers can be designed that support enhanced density of optical modes, without altering the mode density in air, in order to go beyond the 4n2 limit. 

Q13. How many diffraction orders propagate at a given angle?

For a period of 1500 nm (the parameter chosen in Figure 14), 61 diffraction orders propagate at an angle of 8.4 and 62 diffraction orders propagate at an angle of 16.9 with respect to the surface normal. 

Q14. What is the upper bound on the path length enhancement for a single period grating?

This upper bound on the path length enhancement is lower than the Lambertian limit of 4n2, but equal to the 2D enhancement limit for the case of geometrical optics. 

Q15. How can the path length enhancement be maximised?

From Eq. (18), the authors see that the upper bound on the path length enhancement can be maximised by minimising the number of free space propagating diffraction modes, N and maximising the guided optical modes in the absorber, M. The number of free space propagating diffraction modes is minimised by choosing a grating period smaller than or equal to the wavelength of incident light (d k). 

Q16. What is the maximum achievable absorption enhancement for a single period grating?

The maximum achievable absorption enhancement can be increased further by increasing the number of guided optical modes in the absorber, M by using bi-periodic gratings with periodicity equal to the wavelength of incident light.