Broadband optical absorption enhancement through coherent light trapping in thinfilm photovoltaic cells.
ABSTRACT We show that optical absorption in thinfilm photovoltaic cells can be enhanced by inserting a tuned twocomponent aperiodic dielectric stack into the device structure. These coatings are a generalization and unification of the concepts of an antireflection coating used in solar cells and highreflectivity distributed Bragg mirror used in resonant cavityenhanced narrowband photodetectors. Optimized twocomponent coatings approach the physically realizable limit and optimally redistribute the spectral photon densityofstates to enhance the absorption of the active layer across its absorption spectrum. Specific designs for thinfilm organic solar cells increase the photocurrent under AM1.5 illumination, averaged over all incident angles and polarizations, by up to 40%.
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ABSTRACT: Thin semitransparent hydrogenated amorphous silicon (aSi:H) solar cells with selectively transparent and conducting photonic crystal (STCPC) backreflectors are demonstrated. Short circuit current density of a 135 nm thick aSi:H cell with a given STCPC backreflector is enhanced by as much as 23% in comparison to a reference cell with an ITO film functioning as its rear contact. Concurrently, solar irradiance of 295 W/m2 and illuminance of 3480 lux are transmitted through the cell with a given STCPC back reflector under AM1.5 Global tilt illumination, indicating its utility as a source of space heating and lighting, respectively, in building integrated photovoltaic applications.Applied Physics Letters 01/2013; 103(22):2211092211095. · 3.52 Impact Factor  SourceAvailable from: Aaswath Raman
Article: Light trapping in photonic crystals
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ABSTRACT: We consider light trapping in photonic crystals in the weak material absorption limit. Using a rigorous electromagnetic approach, we show that the upper bound on the angleintegrated absorption enhancement by light trapping is proportional to the photonic density of states. The tight bound can be reached if all the states supported by the structure are coupled to external radiation. Numerical simulations are used to illustrate the theory and the design of both two and threedimensional photonic crystals for the purpose of light trapping. Using the van Hove singularities, the angleintegrated absorption enhancement in twodimensional photonic crystals could surpass the conventional limit over substantial bandwidths.Energy & Environmental Science 07/2014; 7(8). · 11.65 Impact Factor  SourceAvailable from: ocean.kisti.re.kr[Show abstract] [Hide abstract]
ABSTRACT: We present the optical models and calculation results of thinfilm organic solar cells (OSCs) at oblique incidence of light, using the transfer matrix method. The simple expression for the optical power dissipation is derived at oblique incidence for s and ppolarized light. The spatial distribution of the electric field intensity, the optical power density, and the optical power dissipation are calculated in both s and ppolarized light with respect to the incidence angle. We identify how the light absorption efficiency for ppolarized light becomes relatively larger than that for spolarized light as the incidence angle increases.Journal of the Optical Society of Korea 03/2012; 16(1). · 1.02 Impact Factor
Page 1
Broadband optical absorption enhancement
through coherent light trapping in thinfilm
photovoltaic cells
Mukul Agrawal and Peter Peumans*
Electrical Engineering, Stanford University, Stanford, CA 94305
*Corresponding author: ppeumans@stanford.edu
http://www.stanford.edu/~ppeumans
Abstract: We show that optical absorption in thinfilm photovoltaic cells
can be enhanced by inserting a tuned twocomponent aperiodic dielectric
stack into the device structure. These coatings are a generalization and
unification of the concepts of an antireflection coating used in solar cells
and highreflectivity distributed Bragg mirror used in resonant cavity
enhanced narrowband photodetectors. Optimized twocomponent coatings
approach the physically realizable limit and optimally redistribute the
spectral photon densityofstates to enhance the absorption of the active
layer across its absorption spectrum. Specific designs for thinfilm organic
solar cells increase the photocurrent under AM1.5 illumination, averaged
over all incident angles and polarizations, by up to 40%.
©2008 Optical Society of America
OCIS codes: (310.6845) Thin film devices and applications; (230.5170) Photodiodes;
(160.4890) Organic materials; (030.1670) Coherent optical effects; (310.4165) Multilayer
design.
References and links
1. S. R. Forrest, “The path to ubiquitous and lowcost organic electronic appliances on plastic,” Nature 428,
911918 (2004).
2. D. Redfield, “Multiplepass thinfilm silicon solar cell,” Appl. Phys. Lett. 25, 647648 (1974).
3. P. Campbell and M. A. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys.
62, 243249 (1987).
4. T. Tiedje, E. Yablonovitch, G. D. Cody and B. G. Brooks, “Limiting efficiency of silicon solar cells,” IEEE
Trans. Electron. Devices ED31, 711716 (1984).
5. P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thinfilm photodetectors and
solar cells,” J. Appl. Phys. 93, 36933723 (2003).
6. L. S. Roman, O. Inganas, T. Granlund, T. Nyberg, M. Svensson, M. R. Andersson and J. C. Hummelen,
“Trapping light in polymer photodiodes with soft embossed gratings,” Adv. Mater. 12, 189195 (2000).
7. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Amer. 72, 899907 (1982).
8. H. Stiebig, N. Senoussaoui, C. Zahren, C. Haase, and J. Muller, “Silicon thinfilm solar cells with
rectangularshaped grating couplers,” Prog. Photovoltaics 14, 1324 (2006).
9. P. Peumans and S. R. Forrest, “Veryhighefficiency doublehetrostructure copper pthalocyanine/C60
photovoltaic cells,” Appl. Phys. Lett. 79, 126128 (2001).
10. P. Peumans, S. Uchida, and S. R. Forrest, “Efficient bulk heterojunction photovoltaic cells using small
molecularweight organic thin films,” Nature 425, 158162 (2003).
11. J. Xue, B. P. Rand, S. Uchida, and S. R. Forrest, “A hybrid planarmixed molecular hetrojunction
photovoltaic cell,” Adv. Mater. 17, 6671 (2005).
12. G. Li, V. Shrotriya, J. Huang, Y. Yao, T. Moriarty, K. Emery, and Y. Yang, “Highefficiency solution
processable polymer photovoltaic cells by selforganization of polymer blends,” Nat. Mater. 4, 864868
(2005).
13. W. Ma, C. Yang, X. Gong, K. Lee, and A. J. Heeger, “Thermally stable, efficient polymer solar cells with
nanoscale control of interpenetrating network morphology,” Adv. Funct. Mater. 15, 16171622 (2005).
14. P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelengthselective absorption enhancement in thinfilm
solar cells,” Appl. Phys. Lett. 43, 579581 (1983).
15. J. Zhao and M. A. Green, “Optimized antireflection coatings for highefficiency silicon solar cells,” IEEE
Trans. Electron. Devices 38, 19251934 (1991).
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16. R. R. Bilyalov, L. Stalmans, L. Schirone, and C. LevyClement, “Use of porous silicon antireflection coating
in multicrystalline silicon solar cell processing,” IEEE Trans. Electron. Devices 46, 20352040 (1999).
17. L. A. A. Pettersson, L. S. Roman, and O. Inganas, “Modeling photocurrent action spectra of photovoltaic
devices based on organic thin film,” J. Appl. Phys. 86, 487496 (1999).
18. K. Kishino, M. S. Unlu, J. Chyi, J. Reed, L. Arsenault, and H. Morkoc, “Resonant cavityenhanced
photodetector,” IEEE J. Quantum Electron. 27, 20252034 (1991).
19. M. S. Unlu and S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607639 (1995).
20. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
21. C. W. Tang, “Two layer organic photovoltaic cell,” Appl. Phys. Lett. 48, 183185 (1986).
22. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1998).
23. M. A. Dupertuis, B. Acklin and M. Proctor, “Generalized energy balance and reciprocity relations for thin
film optics,” J. Opt. Soc. Am. A 11, 11671174 (1994).
24. R. J. Vernon and S. R. Seshadri, “Reflection coefficient and reflected power on a lossy transmission line,”
Proc. IEEE, 57, 101102 (1969).
25. G. P. Ortiz and W. L. Mochan, “Nonadditivity of Poynting vector within opaque media,” J. Opt. Soc. Am. A
22, 28272837 (2005).
26. J. S. Toll, “Causality and the dispersion relation: logical foundations,” Phys. Rev. 104, 17601770 (1956).
27. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, 1972).
28. V. Lucarini, J. J. Saarinen, K.E. Peiponen, and E. M. Vartiainen, KramersKronig Relations in Optical
MaterialsRresearch, (Springer, 2005).
29. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford, 1948).
30. A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, “Phase properties of multilayers,” Appl. Opt. 36,
43824392 (1997).
31. G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical
filters for WDM systems,” IEEE J. Quantum Electron. 34, 13901402 (1998).
32. R. K. Ahrenkiel, “Modified KramersKronig analysis of optical spectra,” J. Opt. Soc. Am, 61, 16511655
(1971).
33. C. A. Emeis, L. J. Oosterhoff, and G. de Vries, “Numerical evaluation of KramersKronig Relations,” Proc.
R. Soc. Lond. A 297, 5465 (1967).
34. K. Tvingstedt, V. Andersson, F. Zhang, and Olle Inganas, “Folded reflective tandem polymer solar cell
doubles efficiency,” Appl. Phys. Lett. 91, 12351411235143 (2007).
35. A. V. Tikhonravov, “Some theoretical aspects of thinfilm optics and their applications,” Appl. Opt. 32,
54175426 (1993).
36. B. T. Sullivan and J. A. Dobrowolski, “Implementation of a numerical needle method for thinfilm design,”
Appl. Opt. 35, 54845492 (1996).
37. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, “Application of the needle optimization technique
to the design of optical coatings,” Appl. Opt. 35, 54935508 (1996).
38. M. Agrawal and P. Peumans, “Design of nonperiodic dielectric stacks for tailoring the emission of organic
lightemitting diodes,” Opt. Express 15, 97159721 (2007).
39. M. Agrawal, Y. Sun, S. R. Forrest, and P. Peumans, “Enhanced outcoupling from organic lightemitting
diodes using aperiodic dielectric mirrors,” Appl. Phys. Lett. 90, 24111212411123 (2007).
40. J. A. Nelder and R. Mead, “Simplex method for function minimization,” Comput. J. 7, 308313 (1965).
41. Matlab 7.0, The Mathworks Inc., Apple Hill Drive, Natick, MA 01760.
42. B. O’Connor, K. H. An, K. Pipe, Y. Zhao, and M. Shtein, “Enhanced optical field intensity distribution in
organic photovoltaic devices using external coatings,” Appl. Phys. Lett. 89, 233502 (2006).
Thinfilm photovoltaic (PV) cells may provide for our energy needs at a price that is
competitive with grid power [1]. However, significant improvements in cost per peak Watt are
still required to reach grid parity. This can be achieved by increasing the efficiency or
reducing the manufacturing cost. The efficiency of a PV cell depends on the likelihood that an
incident photon is absorbed and the efficiency of the subsequent conversion of the photon into
electrical energy. The photon absorption probability improves as the thickness of the active
material is increased. At the same time, the probability that the absorbed energy is collected as
electrical energy at the electrodes reduces as the active layer thickness is increased due to
enhanced recombination [2][6]. Light trapping [2][8] addresses this fundamental tradeoff
by increasing the optical absorption of a given active layer thickness.
A wellknown and effective light trapping approach is to use geometric Lambertian
scattering [7] at one or both surfaces bounding the absorbing layer [Fig. 1(a)]. This approach
increases the absorption by randomizing the occupation of the photon densityofstates
(PDOS) [7]. Since this approach is based on the principles of statistical ray optics [7], it only
applies to PV cells whose active layer thickness is much larger than the wavelength. Thinfilm
PV cells have film thicknesses that are similar to or smaller than the wavelength [8][13]. For
example, in organic PV cells, the optimal active layer thickness is 20nm200nm [9][13]. In
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this case, subwavelength structures can be used to redistribute the PDOS from spectral regions
where absorption is sufficiently strong or not needed to spectral regions where absorption
enhancement would lead to enhanced photocurrents [6][8], [14]. Recently, the use of
embossed gratings to scatter incident light into guided modes in polymer PV cells with active
layer thicknesses <100nm was explored [6]. A maximum increase in external quantum
efficiency of 30% was observed at a wavelength of λ=420nm [6], while the shortcircuit
current under AM1.5 illumination was enhanced by <10%.
Fig. 1. Simplified schematics of thin film PV cells (a) with Lambertian scattering at the
transparent electrode interface, (b) without an ARC, (c) with an ARC over the transparent
electrode, and (d) with a highreflectivity mirror over the transparent electrode in the RCE
configuration.
In this Letter, we evaluate a light trapping scheme for thin film PV cells that uses tailored
dielectric mirrors to build broadband cavities that maximize solar spectral harvesting. Using
the example of an organic PV cell, we show that the short circuit current can be enhanced by
40% for a broadband organic absorber, demonstrating that light trapping methods that rely on
modifying the PDOS have considerable merit for thinfilm PV cells. The dielectric mirrors are
a generalization of the concepts of an antireflection coating (ARC) and distributed Bragg
reflector (DBR).
A simplified schematic of a PV cell in which a semiconductor is sandwiched between a
transparent electrode and a reflecting metal electrode is shown in Fig. 1(b). Due to differences
in refractive index of the different layers, a fraction of the incident light is reflected away from
the device at one of the various entrance interfaces. The remaining light has multiple chances
of being absorbed in the active layer as it makes round trips between the bottom mirror
formed by the reflecting metal electrode and mirror formed by the multiple interfaces between
the absorbing layer and air. Dielectric ARCs [Fig. 1(c)] have found widespread use to reduce
reflection losses in PV cells [15][16]. Since an ARC reduces the airtoabsorbinglayer
reflectivity, it also prevents the optical energy from making multiple round trips inside the
absorbing layer. Even though the probability of multiple round trips reduces, in the limit of
geometric optics (active layer thickness L>>λ), one always benefits from using an ARC.
Summing the multitrip absorption (ignoring optical interference effects) for a structure with
an ideal reflector on the backside (bottom mirror), one obtains that the ratio of the absorption
efficiency of a structure with a perfect ARC over that of one without an ARC is
1 R exp( 2)
(1)
R
−
1
1
L
α−−
. Here, R1 is the reflectivity from air into the active layer without ARC
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and α is the absorption coefficient of the active layer. ARCs always improve the absorption
efficiency when L>>λ even though its utility is lost for weak absorbers (
On the other hand, in thinfilm PV cells, where the thickness of the absorbing layer is
smaller than or similar to the wavelength, optical interference effects are significant and can
be exploited to improve the optical absorption [5], [17]. Highreflectivity dielectric mirrors
[Fig. 1(d)] are commonly used in resonant cavityenhanced (RCE) photodetectors [18][19] to
enhance the absorption over a narrow spectral and angular range using constructive
interference. An approximate but simple analysis [18][19] that neglects the standing wave
effects predicts that the peak absorption enhancement at the resonant wavelength for a
0
L
α→
).
structure with ideal bottom reflector is
1
2
1
(1)
−
(1exp( ))
R
RL
α
−
−
. A more accurate and
detailed analysis is discussed below. The enhancement is evaluated with respect to a structure
bounded by a perfect ARC and ideal reflector. The resonant wavelength is determined by the
resonance condition:
'
N
c
12
2
2
nL
m
ω
φφπ
++=
(1)
where
1
φ and
2 φ are the phase change upon reflection from the top and bottom mirrors, ω
'
N
n is the real part of the refractive index of the active layer, c is the speed
of light, and m is a positive integer. Maximum resonant enhancement of the optical
absorption is achieved when the entrance mirror is “impedance matched” to the
absorber:
1
exp( 2)
RL
α
=−
. In RCE structures, the dielectric mirrors are usually configured
as high reflectivity DBRs such that constructive interference is obtained in the active layer for
a narrow spectral and angular range because the phase of the reflectivity from these mirrors is
correct only for those conditions [18][19].
We now explore whether the concepts of an ARC and mirrors for RCE structures can be
unified to the concept of a mirror that maximizes the optical absorption in a thinfilm solar
cell across its absorption spectrum. Such a mirror needs to exhibit a high reflectivity in the
spectral range where the active layer is a weak absorber to enhance absorption through
constructive interference, and a low reflectivity where the active layer is a strong absorber to
operate as an ARC. In the highreflectivity regions of the spectrum, the mirror needs to have
an anomalous (negative) phase dispersion: the phase upon reflection,
with frequency ω , following Eq. (1). Broadband anomalous phase dispersion would violate
causality, and would mean superluminal group and front velocity [20] and is therefore not
physically realizable. Tradeoffs are therefore necessary.
We first develop a detailed model for two types of organic thinfilm PV cells to evaluate
the dependence of shortcircuit current under AM1.5 illumination on the spectral properties,
i.e. reflectivity,
R ω , and phase,
φ ω , of the top mirror. We then study the upper bound
on the shortcircuit current enhancement imposed by the requirement that the mirror
characteristics are causal. Finally, we provide a practical numerical optimization scheme to
design multilayer dielectric stacks with a performance that approaches the causal limits and
provide two example designs that increase the photocurrent of a thinfilm organic solar cell by
up to 40%.
is the frequency,
1φ , needs to decrease
1( )
1( )
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A schematic of a general thinfilm PV cell on a transparent substrate and with a reflective
back electrode is shown in Fig. 2. Layer N is the active layer for which we want to enhance
the optical absorption. Layer 1 (transparent substrate) and layer M (metal electrode) are
assumed to be semiinfinite. The active (absorbing) layer can be thought of as bounded
between two mirrors. The top mirror is formed by the transparent electrode, sequence of
dielectric layers, and substrate, while the bottom mirror is formed by a spacer layer and metal
electrode. All interfaces are assumed to be optically flat. Light is incident from layer 1 at an
arbitrary angle with mixed s and ppolarization. It is assumed that the absorbing layer
produces electrons and holes. One carrier type is collected by the transparent electrode, while
the other type is transported to the metal electrode via the spacer layer. For our analysis, we
focus on organic donoracceptor (DA) solar cells with an active layer consisting of either a
nanostructured bulk heterojunction (BHJ) of the electron acceptor 3,4,9,10 perylene
tetracarboxylic bisbenzimidazole (PTCBI) and donor copper phthalocyanine (CuPc) [10], or a
CuPc/PTCBI bilayer [21]. The layer structures of the PV cells investigated are glass/100nm
ITO/15nm CuPc:PTCBI (1:1 weight ratio)/20nm bathocuproine (BCP)/Al for the BHJ device
and glass/100nm ITO/11.5nm CuPc/3.5nm PTCBI/20nm BCP/Al for the bilayer device. Note
that very thin active layers (15nm) are used since light trapping is more useful for such
structures. The BCP layer functions as a transparent spacer layer that transports electrons to
the Al cathode [9]. The wavelengthdependent complex refractive indices of CuPc, PTCBI
and Al were measured by spectroscopic ellipsometry. Dispersion in glass, ITO, BCP and the
dielectric coating materials SiO2 and TiO2 was ignored. The refractive indices of all materials
at λ=600nm are tabulated in Table 1.
Fig. 2. Schematic of a thinfilm organic PV cell with an aperiodic dielectric stack between the
glass substrate and transparent ITO anode.
In the bilayer PV cell with the DA interface at
L , the exciton flux at the DA interface as a function of the angle of incidence θ with respect
to the substrate normal can be written as
0
zz
=
and a total active layer thickness of
Layer No. = NLayer No. = N Layer No. = N Layer No. = MLayer No. = M Layer No. = M
AM
AM
AM
AN
AN
AN
B1
B1
B1
BN
BN
BN
BM
rMN
rMN
rMN
XYXYXY
r1N
r1N
r1N
rNM
rNM
rNM
ZZZ
tN1
t1N
t1N
t1N
tMN
tNM
tNM
tNM
LLL
rN1
rN1
rN1
………
A1
A1
A1
GLASSGLASSGLASS
AlAlAl
CuPC:
PTCBI PTCBIPTCBI
r0(N1)1
r0(N1)1
r0(N1)1
t0(N1)1
t0(N1)1
t0(N1)1
Top
MirrorMirror
Bottom
MirrorMirror
Absorbing
LayerLayer
Layer No. = 1Layer No. = 1
ITOITO BCPBCP
BM
BM
tN1
tN1
tMN
tMN
CuPC: CuPC:
Top
Bottom Absorbing
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Table 1. The refractive indices of all materials at λ=600nm in the multilayer BHJ and bilayer cell designs.
( )
θ
( ) ( )
f
θω
∑∫∫
ps
,
∞
=
L
ps exciton
j dzdzzG
0 0
,
,,
ω
(2)
Here
,( , , )
s p
Gz ω θ is the exciton generation rate as a function of location in the active layer,
z, wavelength, λ=2πc/ω, the angle of incidence, θ and the optical polarization (s or p).
( , )
( )
zz
z
′=
′
∂
0
z z
f z
′
∂
=
G
is the exciton collection efficiency, where ( , )
z z′
G
is the Green’s
function for the exciton transport equation
2
2
( , )
z z
dz
′
( , )
z z
τ
()
p
p
Dzz
δ
′′
∂
′
−−=−
GG
with the
boundary conditions
( , )
z z
z
′
∂
Dirac delta function centered at z , and
respectively. Charge carriers are generated by exciton dissociation at the DA interface. It
should be stressed that the Green’s function approach to the solution of exciton diffusion
equation containing a position dependent exciton source term provides significant
computational benefit by avoiding a more commonly used finite difference approach to solve
the inhomogeneous differential equation for each tested mirror structure after evaluating
interference pattern for each such structures.
The BHJ solar cell is also described by Eq. (2) but exciton collection is assumed to be
100% efficient: ( ) 1
f z ≈ . We assume an exponentially decaying probability for the
collection efficiency of charge carriers at the electrodes. The shortcircuit photocurrent
density,
( )( )exp(
exciton
ej
θ
and the carrier collection length,
and processing conditions used.
0
( , )
z z′ =
0
G
(complete quenching at the DA interface) and
0
0
z or L
=
′
∂
=
G
(no exciton quenching at the other interfaces). Here, ()
zz
δ
′−
is the
p
D and
p
τ
are the exciton diffusivity and lifetime,
scj θ , is then equal to
/)
c
L L
−
where e is the electronic charge
cL , is a parameter that depends on the choice of material
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The exciton generation rate,
effects and is given by
,( , , )
s p
Gz ω θ , strongly depends on optical interference
'
N
2
,,
'
1
( )
( )
n
ω
( )
ω
ω
?
( , , )
z
ω θ
( , )
ω θ
( , , )
z
ω θ
N
s p s p
n
GI
ω α
=
E
(3)
where ( , )
I ω θ is incident solar power spectral density per unit projected area as a function
of azimuthal angle, θ.
E
is the relative electric field with respect to the incident
,( , , )
i α are the real part of the refractive index and absorption
coefficient, respectively, of the ith layer. The change in solar power flux as a function of angle
is assumed to follow the cosine law:
( , )
I
ω θ
bottom mirrors as twoport linear systems, one can calculate the electric field inside the
absorbing layer in terms of the twosided scattering coefficients of both mirrors:
+
=
−
ˆ
exp()
( , , )
1
N
ε ω
−
s pz ω θ
electric field and
'
in and
0
( )cos( )
ω
I
θ=
. Modeling the top and the
11
1
exp() exp(2
ik L
)ˆ
( , , )
z
ω θ
1 exp(2)
N zNNM
r
r
N zN
ss
N NM zN
t ik zt ik Lz
r
−
Ee (4)
1,1,
1
ˆ
e
exp(2
ik L
)
exp(2)
N zNp NM
r
r
N zNp
N
p
N NMzN
t ik zt ik Lz
k
z
r
ω θ
+−
+−
=
e
E
(5)
Here,
ˆ
ˆ
e
ˆ
z k
s xy
= ×
ik
+
and
,
ˆ
k
ˆ
k
ˆ
e
()/
pzN xyxyzNN
kkk
±= − ±
are unit vectors and
'
zN
''
zN zN
k
k in the absorbing layer.
k
=
and
xy
ε is the permittivity of the absorbing layer and z is the distance
k are the transverse and inplane components of the wave vector
NN
from the interface of the absorber and the transparent electrode.
Fresnel scattering coefficients [22] as shown in Fig. 2.
Noting that the transmissivities from two sides for any arbitrary layer structure are related
[23] as11
NNz zN
t t kk
=
, Eqs. (2)(5) parameterize the performance of a thinfilm PV cell in

enables us to search for the optimally desired mirror characteristics independent of any
particular physical construction of the mirrors such as layer thicknesses and their refractive
indices. Two of these,
1
and
1
, are constrained through conservation of energy and are
therefore not independent parameters. In thin layers of strong absorbing media, plane waves
are inhomogeneous waves and thus some care is required [24][25] when enforcing energy
conservation relation between
1
and
1
. Instead of using
0
(1)1
Rexp( )
N
r
−
=
deposited on the substrate (see Fig. 2) are more convenient parameters to describe the top
mirror. It should be noted that while
(1)1

N
r
−
about
1

spectral profile of the mirror characteristics. Since the substrate and the dielectrics in the
mirror stack are assumed lossless, one can enforce conservation of energy between
transmissivity
(1)1
N
t
−
and reflectivity
ijr and
ijt are the complex
1/
terms of three optical scattering parameters
1

Nt
,
1
Nr
, and
NM
r
. This parametric model thus
Nt
Nr
Nt
Nr
1
Nr
, we find that the magnitude
and phase of the reflectivity
iφ
of light incident from air on the mirror
0
1
R
=≤ , such an assertion can not be made
Nr
[24][25]. An easily enforced bound on R and φ simplifies locating the optimal
00
N
( 1)1
−
Rexp( )
riφ=
by requiring that
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Page 8
2
0
(
20
N
''
1)1
−
( 1)1
−
01
 (1)/
Nzz
trkk
=−
which then can be propagated [22] to evaluate
1
Nt
and
1
Nr
terms of R and φ .
We first ignore the constraint between R and φ that is imposed by the requirement that
the mirror response be causal and determine for each wavelength, λ, the desired R and φ
that maximize the spectral absorbance given by:
, respectively. Hence, the top mirror (including a lossy anode) can be parameterized in
∫
0
=
L
psps
dzzG
I
,,
),,(
),(
)(
θω
θω
ω
?
ωη
(6)
Fig. 3. (a). Absorbance as a function of wavelength is shown for a device with 15nm thick
active layer (control structure) without a multilayer mirror (dashed line). Also shown are the
spectrallyselective upper bound of absorbance (solid line) and the absorbance of devices with
optimal causal top mirror such that the AM 1.5 weighted average EQE over different target
wavelength bandwidths (BW) is maximized (squares: BW=20nm, circles: BW=60nm,
triangles: BW=150nm, stars: BW=400nm). (b) shows the corresponding amplitude of
reflectance of the top mirror and (c) shows the corresponding phase of reflectance of the top
mirror to achieve the spectrally selectiveupper bound (solid line). The desired amplitude and
phase of reflectance for causal top mirrors designed for broadband (BW=400nm, stars) and
narrowband (BW=20nm, squares) response are also shown.
Because this absorbance can be achieved in a physically realizable device only over a narrow
frequency range, we refer to this as the spectrallyselective upper bound. Figure 3(a) shows
the spectrallyselective upper bound for the spectral absorbance of a CuPc:PTCBI BHJ device
with 15nmthick active layer (solid line) and compares it to that of the unmodified device
structure (dashed line). The desired R (Fig. 3(b), solid line) of the mirror is lowest in the
spectral range where the absorption strength
N
α
possibility of multiple bounces of the light such that a lower R is more desirable to enhance
the incoupling of light by working as an ARC. The ideal φ shows anomalous phase dispersion
across the full spectrum (Fig. 3(c), solid line).
As noted above, this spectrallyselective upperbound cannot be approached over a large
bandwidth. For any physically realizable mirror, the time impulse plane wave reflectivity
)(
1 ) 1(
trN−
)(
1 ) 1(
ω
−
N
r
of the mirror is an analytic function [26][27] in the lower half
L
is largest. Strong absorption eliminates the
~0
must be a causal [26][29] and bounded function [24][25]. This ensures that the
spectral reflectivity
0
of the complex frequency plane while, on the real frequency axis,
)}( Re{
0
N
1 ) 1
−
(
ω
r
and
400500 600700800900
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Spectrally selective upper bound
Control
structure
Wavelength (nm)
150nm BW
Absorbance
400nm BW
20nm BW optimal causal structure
60nm BW
400 500 600 700 800 900
Wavelength (nm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
20nm BW optimal
causal structure
400nm BW
optimal causal
structure
Reflectance
Spectrallyselective upper bound
400 500 600 700 800 900
Wavelength (nm)
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
400nm BW optimal
causal structure
20nm BW optimal
causal structure
Spectrallyselective upper bound
Phase/π
(a) (b)(c)
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)}(Im{
0
N
1 ) 1
−
(
ω
r
r
are related by the generalized dispersion relations [27][28]. In other words,
ω
lie in the upper half complex plane while its zeros can lie in both the
upper and lower halves [30][31]. For broadband enhancement of absorbance, mirrors with
broadband negative phase dispersion are required. Since zeros in the lower half complex plane
contribute to monotonic positive phase dispersion while zeros in the upper half contribute to
the monotonic negative phase dispersion [30][31], a minimumphase system [26],[31] with all
its zeros in the upper half plane is most appropriate for coherent light trapping. For a
minimumphase system,
log())(log(
1 ) 1(
ω
rN
=
−
half space and a generalized dispersion relation known as the singlysubstituted Kramers
Kronig (SSKK) relation with substitution frequency of
))( log(
ω
R
and
)(ωφ
[27][28], [32]:
the poles of
)(
0
N
1 ) 1
−
(
)())(
0
ωφ
i
ω
R
+
is analytic in the lower
1
0
ω =
can be written between
'
'22
0
∫
2
π
log( (
ω
))
( )(0)
R
−
d
ωω
ω
φ ωω φ
′
+
∞
= −
(7)
Here, the longwavelength asymptotic value [28] of reflectivity is taken to be
1
(0)
1
n
+
very long wavelengths. We have developed a perturbative method that starts from a trivial
n
R
n
+
builds another SSKKconsistent pair ( ( )
R ω and ( )
that maximizes the shortcircuit current under AM1.5 illumination in the device geometry
shown in Fig. 2. The kernel of the integral in Eq. (7) is sharply peaked around
Therefore, for each discretized value of frequency, ωj, the values of reflectance R(ωj) at p
neighboring ωj (typically p=57) are simultaneously perturbed within a small range (typically
± 0.01 in q=10 discrete steps). For each of the pq spectral profiles, R(ωj), the corresponding
causal conjugate, φ(ωj), is constructed using Eq. (7). The shortcircuit current under 1 sun
AM1.5 illumination is then evaluated using Eqs. (2)(5) for each of the pq pairs of R(ωj) and
φ(ωj) and the optimal pair is retained. This procedure is repeated for every discretized
frequency ωj. Once a full scan is complete, a 15th degree polynomial is fitted to R(ωj). This
complete procedure is repeated until no further improvements are obtained. The integral in
Eq. (7) is evaluated numerically [33] over a frequency range that is several times larger
(typically 513 times) than the frequency bandwidth over which absorption enhancement is
desired. This reduces the numerical errors inside range of interests and also allows ample
opportunity for the reflectance R(ω) to change outside the range of interest such that both
R(ω) and φ(ω) can approach the desired values in the frequency range of interest. For the
purpose of studying causality imposed upper limit on the achievable solar photovoltaic
conversion through planar coherent light trapping, we restricted ourselves to the case normal
incidence of light.
The optimized SSKKconsistent limits for the absorption of a BHJ device structure when
absorption enhancement is desired over a bandwidth ranging from 20nm to 400nm, is shown
in Fig. 3(a). These results were obtained by optimizing the mirror characteristics subject to the
SSKK for maximum photocurrent of the solar cell as discussed above while limiting the
incident spectrum to a spectral band λ=515nm(515nm+Δ) with Δ=20nm (squares), 60nm
(circles), 150nm (triangles). Also shown is the KKconsistent limit for the full absorption
0
N
1
(1)1
−
1
n
r
−
=
since the optical effects of the multilayer coatings become negligible at
SSKKconsistent combination of
1
1
1
1
( ) 
ω

−
=
and
1
1
1
1
( ) arg()
n
n
φ ω
−
+
=
and gradually
φ ω ) on a discrete frequency grid {ωj},
'
ωω
=
.
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band λ=425nm825nm (stars). The KKconsistent search results in a reflectance amplitude
[Fig. 3(b)] and phase [Fig. 3(c)] profile that is similar to that of a DBR when absorption
enhancement is desired over Δ=20nm (squares). In this case, the enhancement reaches the
spectrallyselective upperbound [Fig. 3(a)]. On the other hand, for PV applications, where
broadband absorption enhancement is desired, negative phase dispersion is required as seen in
Fig. 3(c) (stars) and it is not possible to reach the spectrallyselective upper bound and there
is a clear tradeoff between spectral bandwidth and achievable enhancement. These
spectra represent the theoretical limit of photocurrent enhancement that can be achieved using
the planar coherent lighttrapping technique that can be physically implemented using discrete
refractive index multilayer coatings or as planar graded index structures.
Figure 4(a) shows the achievable average AM 1.5weighted EQE for the BHJ device using
mirrors that satisfy the SSKKconstraint for the full absorption band as a function of the active
layer thickness (solid triangles) for critical lengths of charge collection Lc=20nm and 40nm.
The average EQE for a control device without dielectric mirror is also shown for reference
(open triangles). Despite the limits imposed by Eq. (7), optical interference effects can be used
to trap light and improve the photocurrent produced by a thinfilm solar cell. For the 15nm
thick BHJ device, the model predicts an increase in shortcircuit current density from
3.2mA/cm2 for the control device to 4.9mA/cm2 when the SSKKconsistent mirror is
included, representing a 53% increase. The achievable enhancement in AM1.5weighted
average EQE as a function of active layer thickness is also shown (circles). As expected, the
thinnest device structures benefit most from light trapping.
Fig. 4. (a). AM1.5weighted average EQE for the control BHJ device (open triangles) and a
BHJ device with an optimal causal top mirror (filled triangles) for different charge collection
lengths (LC=20nm and 40nm) as a function of the thickness of active layer. Also shown is the
broadband improvement achieved through coherent light trapping as a function of the active
layer thickness. (b). Optimal causal limit of absorbance averaged over the spectral range
λ=425nm825nm that is achievable as a function of the targeted bandwidth for a device with
15nmthick active layer. Inset: Example of an absorbance spectrum without stack (open
squares) and with stack (solid squares) optimized for a bandwidth of 150nm.
Figure 4(b) shows the average absorption (averaged for the spectral range λ=425nm
825nm) as a function of the bandwidth over which absorption enhancement is desired for the
BHJ device with a 15nmthick active layer. Structures with a tuned mirror that absorb ~56%
of the incident power within a 150nm bandwidth are physically realizable [see inset to Fig.
4(b)], despite the mirrorless structure only absorbing 28% over the same bandwidth. While
150nm is narrow compared to the solar spectrum, the enhancement of the absorption in well
defined spectral bands might be useful in tandem configurations where separate cells are used
for different spectral regions [34].
We now present a practical numerical optimization scheme to design twocomponent
dielectric stacks with characteristics that approach the limit set by the SSKK relation. It has
been shown that twocomponent coatings with the maximum and minimum refractive indices
0 2040 60 80100 120 140 160
0
5
10
15
20
25
30
0
10
20
30
40
50
60
Without stack
With stack
Improvement
LC=20nm
AM1.5 Weighted Average EQE (%)
Active Layer Thickness (nm)
LC=40nm
AM1.5 Weighted Improvement (%)
0 50 100 150 200 250 300 350 400 450
Target Bandwidth (nm)
10
20
30
40
50
60
70
Average Absorbance (%)
400 500
Wavelength (nm)
600 700800900
0
10
20
30
40
50
60
70
80
90
100
With stack
Full BW 400nm
(515+Δ Δ)nm
515nm
825nm
Absorbance (%)
425nm
Target bandwidth
Without stack
(a) (b)
Average absorption without stack
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available are optimal coatings [35]. The stacks use SiO2 (n=1.5) and TiO2 (n=2.5) as the low
and highindex material, respectively. The optimization procedure targets to achieve
maximum photocurrent averaged over s and p polarizations and over a whole 12hour day.
The spectrum is assumed to be the AM1.5 spectrum whose intensity varies as the cosine of the
incident angle. The optimization routine starts with a thinfilm organic PV cell with a single
thick dielectric layer of SiO2 inserted between the glass substrate and ITO electrode. An
exhaustive search is then performed for the optimal location for the insertion of a new layer
[36][39]. Since characteristic matrices are stored, this exhaustive search only requires the
calculation of three additional characteristic matrices for each tested structure. Repeated
insertion of layers is performed before a NelderMead simplex algorithm [40] is used to find a
locally optimal set of layer thicknesses. The design is automatically purged of any layers less
than 6nm thick. This procedure is repeated with different thicknesses and different numbers of
simultaneous new layer insertions. The optimization ends when the desired performance is
achieved or if no further layers can be increased without degrading performance. If the desired
performance is not achieved, the overall coating thickness may be increased to enable further
layer insertions. A similar optimization procedure was previously developed [38] and
experimentally verified [39] by the authors for applications in organic light emitting diodes.
The present optimization scheme was implemented in a commercial linear algebra package
[41]. The results of design procedure are a 4layer design for the BHJ cell and a 6layer design
for the bilayer cell. The resulting stack designs are shown in Table 1.
35
Bilayer no stack
Bilayer with stack
BHJ no stack
BHJ with Stack
Fig. 5. (a). External quantum efficiency for the bilayer (squares) and the BHJ (triangles) 15nm
thick cells with (filled symbols) and without (open symbols) an optimized dielectric stack
between the substrate and ITO layer. (b) Normalized shortcircuit current as a function of angle
of incidence of the illumination for a 15nmthick bilayer device with (filled squares) and
without (open squares) an optimized dielectric stack. The short circuit current at normal
incidence is normalized to 1. The enhancement in shortcircuit current density by insertion of
the optimized dielectric stack is also shown for the bilayer (circles) and the BHJ (triangles)
devices.
Figure 5(a) shows the modeled external quantum efficiency at normal incidence for the
bilayer (squares) and the BHJ (triangles) cell with (solid symbols) and without (open symbols)
the dielectric mirrors. The charge collection efficiency was assumed to be 100% since we are
interested in the achievable enhancement. The optimization scheme resulted in a decrease in
response at short wavelengths and an enhancement in response for longer wavelengths to
match the cell response to the solar spectrum. This is an example of a redistribution of the
PDOS to achieve better performance. The shortcircuit current density under AM1.5
illumination increases from 2.6mA/cm2 to 3.7mA/cm2 for the bilayer cell (a 42%
enhancement), and from 3.2mA/cm2 to 4.3mA/cm2 for the BHJ cell (a 34% enhancement).
The effects described here were observed in a simplified structure where a single dielectric
layer was used to increase the monochromatic photocurrent in an organic thinfilm solar cell
400 450 500 550 600 650 700 750 800 850
Wavelength (nm)
0
5
10
15
20
25
30
External Quantum Efficiency (%)
0 1020 30
Angle (degrees)
40 50 607080 90
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Photocurrent
Normalized current
Bilayer no stack
Bilayer with stack
Improvement
Bilayer
BHJ
(b)
cos(θ)
(a)
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by 30% [42]. The methods presented here provide a larger and spectrally broader
enhancement.
Figure 5(b) shows the dependence of normalized shortcircuit current density on the angle
of incidence, θ , for the bilayer cells with (filled squares) and without (open squares) an
optimized dielectric stack. The shortcircuit current density is normalized to unity at normal
incidence. The interference coating does not result in a strong dependence of shortcircuit
current density on θ. Both with and without dielectric stack, short circuit current
approximately follows a cos(θ)dependence because the optical flux varies as cos(θ) (grey
line). The enhancement achieved in the shortcircuit current density as a function of azimuthal
angle by inclusion of the dielectric stacks for the bilayer (filled circles) and the BHJ (filled
triangles) cell, is also shown. At normal incidence, a 42% and 34% improvement in short
circuit current density is obtained for the bilayer and BHJ cell, respectively. For angles >50°
the enhancement decreases rapidly. The optimization procedure automatically optimized for
angles near normal incidence since the optical flux falls off as cos(θ). The enhancement in
photocurrent averaged over a full day is 40% and 33% for the bilayer and BHJ cell,
respectively.
In conclusion, we have shown that specifically designed dielectric mirrors can be used to
substantially enhance the optical absorption in thinfilm PV cells by modifying the PDOS.
These dielectric mirrors are a generalization of ARCs and mirrors for RCE photodetectors.
Using thinfilm organic PV cell model systems, we have shown that enhancements in
photocurrent of up to 40% are achievable despite a broadband absorption spectrum. With low
cost techniques to manufacture such thinfilm dielectric coatings, this approach may be of
practical use to enhance the efficiency of thinfilm solar cells.
Acknowledgment
This work was partially supported by the Stanford Global Climate and Energy Project and
AFOSR #FA95500610399.
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