Page 1

Bridging electromagnetic and carrier transport

calculations for three-dimensional modelling of

plasmonic solar cells

Xiaofeng Li,* Nicholas P. Hylton, Vincenzo Giannini, Kan-Hua Lee, Ned J. Ekins-

Daukes, and Stefan A. Maier

Blackett Laboratory, Department of Physics, Imperial College London, London SW7 2AZ, UK

*xfl_79@yahoo.com.cn

Abstract: We report three-dimensional modelling of plasmonic solar cells

in which electromagnetic simulation is directly linked to carrier transport

calculations. To date, descriptions of plasmonic solar cells have only

involved electromagnetic modelling without realistic assumptions about

carrier transport, and we found that this leads to considerable discrepancies

in behaviour particularly for devices based on materials with low carrier

mobility. Enhanced light absorption and improved electronic response

arising from plasmonic nanoparticle arrays on the solar cell surface are

observed, in good agreement with previous experiments. The complete

three-dimensional modelling provides a means to design plasmonic solar

cells accurately with a thorough understanding of the plasmonic interaction

with a photovoltaic device.

© 2011 Optical Society of America

OCIS codes: (040.5350) Photovoltaic; (240.6680) Surface plasmon; (290.1990) Diffusion.

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1. Introduction

The application of surface plasmons (SPs) in photovoltaic devices has recently attracted

considerable attention due to potential improvements to device absorption without increasing

the physical thickness of the photoactive medium [1–8]. Performance improvement has been

predicted through simulations and verified in experiments for solar cells (SCs) based on both

inorganic and organic materials [9–15] with properly designed plasmonic nanostructures [16–

25]. In order to comprehensively evaluate a SC, several crucial device parameters, e.g., short-

circuit photocurrent density (Jsc), current-voltage (I-V) curve, fill factor (FF), light-conversion

efficiency (η), etc. are required. These parameters can be readily achieved in photovoltaic

experiments [4,10,12,25]; however, most previous simulations of plasmonic SCs (PSCs)

consider only the optical absorption through solving Maxwell’s equations. Although improved

photocurrent has been predicted in several articles [17,21], ideal carrier transport (ICT) was

assumed, namely that each photo-generated electron-hole pair can successfully contribute to

the photocurrent. Under this condition the realistic solar cell behaviour and configuration

(e.g., doping profile, carrier mobility, carrier loss, etc.) are ignored. However, this approach is

not always accurate since in a real SC the photo-generated carriers will recombine (prior to

collection) under various loss mechanisms, such as surface and bulk carrier recombination or

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short carrier diffusion [26]. The device performance is overestimated especially for those SCs

based on materials with low carrier mobility and short carrier lifetime, e.g., hydrogenated

amorphous silicon (α-Si:H). Hence, the exact electronic response must be considered in order

to obtain accurate I-V characteristics for a complete evaluation of the SCs. Therefore, an

accurate model of PSCs must consider both the electromagnetic (light absorption) and

electronic (carrier transport) properties together in a single simulation. Due to the strongly

localized confinement of SP waves, such work has to be performed in both frequency and all

spatial domains.

This paper presents exact three-dimensional (3D) simulations of PSCs that link both

optical absorption and carrier transport together. We have verified the need to perform the full

calculation of the electronic response of a SC by comparing the results of gallium arsenide

(GaAs) and a-Si:H SCs under a purely electromagnetic treatment and using our full

calculation. External quantum efficiency (EQE) and I-V characteristics before and after

plasmonic design were then calculated, in order to obtain complete information about the

performance improvement of SCs (e.g., Jsc, FF, η, etc). Finally, the spatial distributions of

power flow and carrier concentration profiles were extracted from the simulation.

2. Theory

Our work aims to realize a modular 2-step 3D simulation based on the finite element method

[27]. The first step is to perform an exact electromagnetic calculation (inherent in which are

all optical mechanisms, including any shading effect due to metallic nanoparticles on the

surface) based on 3D Maxwell’s equations in the frequency domain. A single unit cell was

used in conjunction with periodic Floquet-Bloch boundaries to represent the whole structure,

and the standard AM1.5G spectrum was introduced as the incident light source. Linearly

polarized light was used since the device is rotationally symmetric and insensitive to the light

polarization. The photo-generation rate extracted from the optical calculation is also used for

the exact carrier transport calculation in the second step, which is based on two sub-modules

simulating electron and hole transports, respectively, and one sub-module for electrostatic

potential (see the supplemental material). The 3D transport equations were solved under solar

injection and forward electric bias for the calculation of photocurrent and dark current.

Finally, information on SC performance, including EQE, Jsc, I-V curve, open-circuit voltage

(Voc), maximum output power density (Pmax), FF, η, etc. can all be obtained for the

optimisation of PSC designs.

The electronic response of solar cells (SCs) can be simulated accurately using three-

dimensional (3D) transport equations for electrons and holes. In order to develop a general

model capable of simulating various SCs, including those based on heterojunctions,

discontinuity of the intrinsic material parameters at heterojunction interfaces must be taken

into account. The simulation of homojunction SCs is then a specific case where the

discontinuity term is removed from the carrier transport equations. The generalized 3D carrier

transport equations (in steady state) used in our model are [26]

ln( , , , )

g x y z

,

B

q

nnc

k T

DnnNU

q

(1)

ln( , , , )

g x y z

,

g

B

q

ppv

E

q

k T

DppNU

q

(2)

where n (p) is the electron (hole) concentration, Dn = µnKBT/q (Dp = µpKBT/q) is the electron

(hole) diffusion coefficient, µn (µp) is the electron (hole) mobility, KB is Boltzmann’s

constant, T ( = 300 K) is the operating temperature, q is the electron charge, Ф is the

electrostatic potential, χ is the electron affinity, Eg is the material band gap, Nc (Nv) is the

effective conduction (valence) band density of states, g(x, y, z, λ) = α(x, y, z, λ)Ps(x, y, z, λ) is

the generation rate, α is the material absorption coefficient, Ps is the power flux calculated

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from the electromagnetic model, and U is the rate of carrier loss due to Shockley-Read-Hall,

radiative, and Auger recombination.

The electrostatic potential Ф is determined by the charge profiles in the device according

to the 3D Poisson’s equation

2

,

qn

p C

(3)

where ε is the material permittivity, C = Nd – Na is the impurity concentration defined as the

sum of the concentrations of the ionized donors Nd and acceptors Na, including the signs of the

compensated charges.

The calculated spatial profiles of n and p are used to compute the frequency-dependent

photocurrent for electrons and holes, respectively [i.e., jn(x,y,z,λ) = qDnn and jp(x,y,z,λ) = –

qDpp]. The corresponding short-circuit current density jsc(λ) is then given by the averaged

photocurrent at the surface of the SCs

/2 /2

2

/2/2

1

( )

( , , , )

j x y L

( , , , )

x y L

,

scnp

jj dxdy

(4)

where L is the total thickness of the photoactive layers. The external quantum efficiency

(EQE) is then obtained by using EQE(λ) = jsc(λ)/[qbs(λ)], where bs is the solar incident flux.

The overall short-circuit current density Jsc can then be expressed by Jsc = ʃ jsc(λ)dλ.

Considering dark current Jd(V) and device resistances, the current-voltage (I-V) relation is

written as

( )

J V R

R

( )

J V

( ),

s

scd

sh

V

J J V

(5)

where Rs and Rsh are series and shunt resistances, respectively, and Jd(V) is obtained by

applying forward electric bias into Eqs. (1) and (2), neglecting the generation terms. From the

I-V curve, parameters including open-circuit voltage Voc, maximum output power density

Pmax, fill factor [FF = Pmax/(JscVoc)], and light-conversion efficiency (η = Pmax/Psun, where Psun

is the overall incident light power density) can be obtained.

3. Simulation results

Table 1. Key Parameters Used in the Simulation [23,25–29].

Layers

window

emitter

base

BSF

p region

i region

n region

Doping

(cm–3)

1 × 1018

4 × 1018

2 × 1017

2 × 1018

1.3 × 1017

µn

(cm2/Vs)

100

1100

4000

100

4.6 × 102

4.6 × 102

1

µp

(cm2/Vs)

10

80

80

10

0.5

9.2 × 103

9.2 × 103

τn

τp

GaAs SCs

1 ps

1 ns

1 ns

1 ns

2 µs

2 µs

1.7 µs

10 ns

10 ns

10 ns

1 ps

0.34 µs

2 µs

2 µs

α-Si:H

SCs

4.3 × 1016

BSF: back surface field. The doping concentration shown here for α-Si:H SCs is the ionised

donor/acceptor concentration; actual dopant levels can be much higher.

Listed in Table 1 [26,28–32] are the key parameters employed in the carrier transport

calculations. Additionally, the surface recombination coefficients for minority electrons and

holes are taken to be 1 × 106 (1 × 104) cm/s for GaAs (α-Si:H) SCs. Typical non-plasmonic

GaAs- and α-Si:H-based SCs [shown in Figs. 1(a) and 1(b) respectively] were first studied to

validate the necessity of performing the complete electronic calculation for a real SC. We

consider these two types of SCs since they are based on direct-band photoactive materials

with typically high and low carrier mobility, respectively, and can therefore test the two

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modelling treatments under different conditions. Moreover, the large absorption coefficients

of the materials make them well-suited to thin-film PSC applications.

Simulation results are displayed in Figs. 1(c) – 1(f), where the results for “full model”

originate from the complete 3D calculation, while those for ICT were predicted by spatially

integrating and surface-averaging the carrier generation rate (multiplied by electron charge)

obtained from the optical calculation. Both current density and EQE of the SCs under solar

illumination were analyzed and a discrepancy between the estimation limited to

electromagnetic modelling and complete device modelling was observed. This discrepancy

relates to the device’s internal quantum efficiency (IQE), which can be directly approximated

by the ratio of the photocurrents derived from the exact calculation and the assumption of

ICT. For GaAs SCs, photocurrent and EQE are overestimated in the short-wavelength part of

the spectrum [Fig. 1(e)], while for α-Si:H SCs distinct overestimation was found over a much

broader spectral range [Fig. 1(f)]. We believe that the photocurrent decrease in real GaAs SCs

(compared to that estimated optically) primarily originates from surface recombination, which

is more pronounced at short wavelengths where the absorption coefficient increases

dramatically and a large fraction of the incident light is absorbed close to the surface.

However, α-Si:H SCs suffer from carrier loss due not only to recombination in the thin emitter

(including) but also very low carrier mobility. The latter occurs over a broad spectral range,

leading to a low IQE across much of the absorption spectrum. According to our calculation,

the exact value of Jsc under complete 3D calculation is 17.16 (9.15) mA/cm2 for the GaAs (α-

Si:H) SCs and that estimated from optical calculation is 17.93 (10.88) mA/cm2. The

overestimated value is noticeable and impacts the PSC design (discussed later). It should be

noted that the accuracy of pure optical treatment strongly depends on the device

configuration; therefore, a robust model simulating both optical and electronic response is

necessary.

Fig. 1. (a) and (b): schematic diagrams of the considered GaAs- and α-Si:H SCs; (c) and (d):

current densities from solar incidence and those generated from the SCs; (e) and (f): EQE and

IQE response of the SCs. (a), (c) and (e) [(b), (d) and (f)] are for GaAs (α-Si:H) SCs. The

results obtained under ICT assumption are plotted with dashed curves. The observed peaks

between 550 nm and 800 nm for α-Si:H SCs are due to Fabry-Perot interference in the cavity.

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Plasmonic effects can also be incorporated into our model for the simulation of PSC

structures [see Figs. 2(a) and 2(b)]. As an example, SCs were simulated with periodic arrays

of silver nanoparticles (with a diameter of 160 nm and 400 nm period) on the top surface. The

effect of plasmonic design on EQE response is displayed in Figs. 2(c) and 2(d), where both

EQE (i.e., photocurrent) degradation and enhancement can be observed. This is a typical

feature as reported in previous PSC experiments [12,25,33]. EQE decrease/increase in PSCs

was believed to be the result of the destructive/constructive interference between the scattered

field from the nanoparticles and that transmitted directly from the incidence, occurring on the

blue/red sides of the plasmonic resonance [33]. Alternatively, a recent explanation for this

phenomenon suggests that the photocurrent decrease at short wavelengths is due to the

resonant modes at the top of the particles, while the enhancement at long wavelengths is from

the modes localized at the Ag/substrate interface [34]. We have indeed observed similar

behaviour (not shown here) by checking the electric field distribution calculated from our

model. Note also that the spectra in Figs. 2(c) and 2(d) exhibit a sharp feature at a wavelength

of 400 nm, which is due to the first diffracted order arising from the periodicity of the

nanoparticles.

Under the proposed plasmonic design EQE enhancement was realized when λ > 455 nm

(447 nm) for GaAs (α-Si:H) PSCs, benefiting from the preferentially forward light scattering

and strong near-field confinement of SPs [1–4]. The plasmonic effect can be controlled

effectively by tuning the resonance of a single particle and the coupling strength of the

nanoparticles; therefore, the performance of PSCs can be optimized. On the other hand,

without considering carrier transport, as shown in Fig. 2, EQE is overestimated and the critical

wavelength distinguishing the photocurrent loss and gain regions is blue-shifted.

Fig. 2. Schematic diagram (a) [(b)], EQE response (c) [(d)] and I-V curves (e) [(f)] of GaAs [α-

Si:H] SCs before (dashed) and after (solid) the proposed plasmonic design with 160nm-

diameter silver particles under period of 400 nm. Solid and dot curves are from complete

calculation and optical estimation under ICT assumption, respectively. Power densities are also

plotted in (e) and (f) so that the detailed information about Jsc, Voc, Pmax, FF, and η can be

obtained. The performance enhancement is observed from these figures and the extracted

performance parameters are listed in Table 2.

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In order to acquire the I-V information, the carrier transport module is used again to

calculate the dark current response of the SCs under forwardly applied voltages. This was

realized in our model by controlling the boundary conditions of the electrostatic potential sub-

module [26]; the results are plotted in Figs. 2(e) and 2(f), respectively. In addition, output

power densities have also been plotted so that Pmax, FF, and η could be determined. From

these figures, the device performance parameters under various conditions were extracted and

listed in Table 2, which shows that, with the incorporation of plasmonic nanostructures,

improvement ratios in Jsc, Pmax, and η of more than 18% for GaAs SCs and over 40% for α-

Si:H SCs could be achieved. As expected, over-predicted performance under ICT assumption

was observed. The difference between these two methods is especially distinct for α-Si:H SCs

as shown in Fig. 2(f) and Table 2.

Table 2. Performance Comparison under Various System Configurations

Jsc

(mA/cm2)

17.16

20.28

(18.2%)

21.00

(22.4%)

9.15

12.78

(39.7%)

15.12

(65.3%)

Voc

(V)

1.028

1.034

(0.58%)

1.035

(0.68%)

0.94

0.95

(1.1%)

0.96

(2.1%)

Pmax

(mW/cm2)

14.34

17.10

(19.3%)

17.73

(23.6%)

7.18

10.20

(42.1%)

12.14

(69.1%)

FF

81.3%

81.6%

(0.37%)

81.6%

(0.37%)

83.5%

84.0%

(0.6%)

83.6%

(0.12%)

η

GaAs SCs

no SP

14.9%

17.8%

(19.5%)

18.4%

(23.5%)

7.5%

10.6%

(41.3%)

12.16%

(68.0%)

PSC (full model)

(enhancement, %)

PSC (ICT)

(enhancement, %)

no SP

PSC (full model)

(enhancement, %)

PSC (ICT)

(enhancement, %)

α-Si:H SCs

However, neither Voc nor FF exhibit noticeable increases [11,35]. Note that the FF shown

here is relatively high, since ideal series (i.e., 0 Ωcm2) and shunt resistances (i.e., Ωcm2)

were used for simplicity. Realistic resistance values could be easily integrated into our model

[see Eq. (5)], resulting in a more credible FF. Note also that the incorporation of metallic

components on the incident surface may lead to an associated reduction in resistance, hence

increasing the FF [12,26]. This effect can be included by using appropriate device resistances

that reflect the effect of a metallic nanoparticle coating. We also emphasize that the scope of

our work is to obtain a full 3D model that considers both the electromagnetic and carrier

transport response, and that the optimization of PSCs is beyond the scope of this paper; hence

the device performance shown herein may not be maximized compared to some previous

reports (e.g., for α-Si:H SCs reported in [36]).

We would like to indicate that there also exists a compromise between a full 3D model

and the ICT case by using only low-dimensional electronic model, i.e., based on the

assumption that the IQE of the original SC shows no noticeable change after the introduction

of plasmonic nanoparticles. In this compromise the original IQE from a 1D electronic

calculation and the field profile of the PSCs from our 3D electromagnetic calculation are used

to predict the final electronic output. Using this method, Jsc values for the considered GaAs

and α-Si:H PSCs were recalculated and found to be 20.25 and 13.00 mA/cm2, respectively,

showing an obvious improvement from the ICT case. However, it is unclear if such a

compromise remains valid for all cases (different materials and system setups or strong

plasmonic effects). For example, if the solar cells are under multi-sun illumination or a higher

nanoparticle concentration is employed, the carrier profiles will be modified more strongly by

plasmonic design leading to significant modification of the IQE. Furthermore, obtaining

detailed spatial information of the electronic parameters is also an important issue, which can

only be obtained from a complete 3D simulation.

Therefore, we now turn our attention to the benefits of our 3D simulation in enabling the

extraction of spatial information on optical and electronic parameters. Shown in Fig. 3 are the

power flows in the α-Si:H PSCs (GaAs PSCs show qualitatively same behaviour) working in

the photocurrent loss [λ = 350 nm, Figs. 3(a1) and 3(a2)] and gain [λ = 500 nm, Figs. 3(b1) –

3(b3)] regions, respectively. The images shown in Fig. 3(b3) correspond to the various z

(C) 2011 OSA 4 July 2011 / Vol. 19, No. S4 / OPTICS EXPRESS A894

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positions shown in Fig. 3(b1). The figure clearly shows that the uniform distribution in the xy

plane and exponential decay along the z direction in a conventional non-plasmonic SC have

been strongly modified due to the presence of plasmonic nanostructures. Power confined in

the active layers can be either decreased under short-wavelength incidence [see Figs. 3(a1)

and 3(a2)] or increased under long-wavelength incidence [see Figs. 3(b1) – 3(b3)]. In the

photocurrent gain region, the incident light can undergo strong forward scattering and

confinement in the photoactive regions. This is especially beneficial to increase the effective

optical absorption length so that photocurrent can be generated more efficiently.

Fig. 3. Power flow distributions in the plasmonic α-Si:H SCs with 160nm-diameter silver

nanoparticles decorated above. In (a1) and (a2), λ = 350 nm (in photocurrent loss region) and in

(b1), (b2), and (b3) λ = 500 nm (in photocurrent gain region). In (b3), field polarizations are

also given.

Figure 4 illustrates the calculated distributions of carrier generation rate and electron and

hole concentrations inside the active layers of α-Si:H SCs working at λ = 500 nm [in the

region of photocurrent gain as shown in Fig. 2(b)]. It is clear that the spatially modulated

incident wave shown in Fig. 3 reshapes the carrier generation profile as well as modifies the

carrier concentration distribution. However, the overall carrier concentration profiles are not

significantly changed after solar injection (not shown here). This is firstly because the photo-

generated carrier concentration under one-sun illumination is far less than the background;

secondly, the spatial non-uniformity will be alleviated with carrier transporting inside the

active layers.

The transverse dependence of the stabilized carrier profiles arising from the plasmonic

nanostructures can be best seen on a logarithmic scale, as shown by Figs. 4(b1) – 4(c2). In

addition, spatial information of other electronic properties (e.g. electrostatic potential,

electrostatic field, carrier recombination rate, etc.) can also be obtained from our simulation.

Here, we would like to emphasize that the ability to obtain detailed carrier concentration

distributions inside the device is an important and experimentally accessible value. For

example, a terahertz near-field microscopy technique was recently reported [37], where the

authors demonstrated that the technique is capable of measuring carrier densities close to the

surface of a material. They showed that regions with metals or highly conductive

(C) 2011 OSA4 July 2011 / Vol. 19, No. S4 / OPTICS EXPRESS A895

#145911 - $15.00 USD Received 13 Apr 2011; revised 11 May 2011; accepted 13 May 2011; published 30 Jun 2011

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semiconductors appear brightest in the THz image while semiconductors with low doping

levels appear darker. Our model can provide a powerful theoretical tool to aid interpretation or

corroborate the results of the terahertz near-field microscopy technique. Our model is also

able to calculate the 3D carrier distribution throughout the materials, not just at the surface but

internally. Since carrier recombination proceeds via different channels as various powers of

carrier density, it is critical to account for local variation in carrier density to accurately

predict photovoltaic device performance.

Fig. 4. Distributions of carrier generation rate [(a1) & (a2)], electron concentration [(b1) &

(b2)], and hole concentration [(c1) & (c2)] inside the active layers of α-Si:H PSCs working at λ

= 500 nm (in photocurrent gain region). Active layer configuration has been shown in Figs.

1(b) and 3(a1).

4. Conclusion

In summary, PSCs have been simulated using our 3D model, which calculates both the optical

and electronic responses of a solar cell structure. Besides the extensively investigated optical

properties, it also provides 1) a realistic measurement of the electronic performance

enhancements arising from the presence of metallic nanoparticles and 2) spatial distributions

of electronic parameters inside the device. Comparison between complete device simulation

and optical estimation shows that carrier transport needs to be considered in a realistic SC

model. Typical findings agreeing with PSC experiments and performance improvement with

the incorporation of plasmonic design are verified. Since this model limits neither plasmonic

nanostructure geometry nor SC configuration, it can be widely applied for the modelling of

PSCs and conventional non-plasmonic SCs in 3D. Minor modifications to the model would

enable light-emitting diodes and photodetectors to be modelled in a similar way.

Acknowledgments

This work was supported by EU FP7 project “PRIMA” - 248154.

(C) 2011 OSA4 July 2011 / Vol. 19, No. S4 / OPTICS EXPRESS A896

#145911 - $15.00 USD Received 13 Apr 2011; revised 11 May 2011; accepted 13 May 2011; published 30 Jun 2011