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On the efficiency of solar energy harvesting with

nano-dipoles

Zhongkun Ma1, Xuezhi Zheng1, Guy A. E. Vandenbosch1, V. V. Moshchalkov21

Dep. Electrical Eng,. KU Leuven, Leuven, Belgium, [email protected]

INPAC, KU Leuven, Leuven, Belgium

AbstractNano-antennas or nantennas are considered as a possiblealternative to conventional solar cells. The radiation efficiency of a

nantenna is the first factor in the total efficiency product by which these

systems are able to convert incident light into useful energy. This

efficiency is investigated in terms of the metal used as conductor and the

dimensions of the nano-antenna. The results define upper bounds for any

possible process transforming light into electrical energy. Silver shows the

highest efficiencies, with radiation efficiencies near or slightly above 90

%, and a total solar power harvesting efficiency of about 60 70 %. This

is considerably higher than conventional solar cells. It is found that fine-tuning of the dipole dimensions is crucial to optimize the efficiency.

Index TermsSolar energy harvesting, solar radiation, nano

antennas, nano dipoles

I. INTRODUCTIONSolar energy is expected to deliver a large contribution to

the solution of human kinds energy problem. At this moment,

90% of the solar cells sold are crystalline silicon wafers. The

disadvantage of this technology is the lower efficiency.

Typical efficiencies are in the order of 20-30%.

In recent years, the idea of using nano-rectennas (nano-

antenna or nantenna + rectifier) to harvest solar energy hasbeen suggested. It is claimed that the efficiency of this type of

topology may be much larger. The figures mentioned go from

a staggering 90% [1], to a more down-to-earth 30-40 % [2].

In this paper realistic numbers are presented for antenna

efficiencies that can be reached with nano-antenna technology.

These numbers are based on a detailed study of a singleantenna topology, the basic dipole, for a range of different

metals and different sizes.

The total efficiency of nano-rectennas consists of two parts.

The first part is the efficiency by which the light is captured

by the nano-antenna and brought to its terminals. Due to

reciprocity, this efficiency is the same as the efficiency by

which the antenna is able to convert input power given at itsterminals into radiation. This efficiency is thus the radiation

efficiencyrad of the antenna. Although this efficiency has

been very well studied for traditional antennas, the in-depth

characterization of this parameter has not yet been addressed

in the nano-antenna research community. To start with, in by

far most papers on nano-antennas known to the authors, only

gold is considered as metal. Concerning topologies, some

information can be found [3], [4]. However, Gao [3] considers

only two structures of the same length and a very rough Drude

model is used in the FDTD solver used, fitting the

experimental material parameter data. It can be proven that

this affects the efficiencies considerably. Huang [4] uses only

a single frequency.

The second part is the efficiency by which the captured

light is transformed into low frequency electrical power by the

rectifier. At lower frequencies, rectifying circuits are common,

but at IR and optical frequencies and in combination with

nano-antennas, efficient rectification is a real challenge, see

[5] where an efficiency is reached of 0.01 %. This very low

figure is in sharp contrast with the efficiencies mentioned by

Kotter [1] and Service [2], and it illustrates the long way still

to go before real practical use can be made of solar energy

harvesting with nano-antennas.

This paper considers the first step only, the capturing of the

IR and optical waves and the transport of the energy

embedded in these waves to the terminals of the nano-antenna.

It is essential to emphasize that the interaction between light

and nano-antennas in the frequency bands considered can still

be analyzed with a high degree of accuracy using classical

electromagnetic theory [3], [6], [7]. The fact that at this small

scale, no quantum effects have to be taken into account isreally a crucial observation. It means that the concept of an

antenna, a device able to transmit and receive

electromagnetic waves rather than particles, still works..The main target of this paper is to derive upper bounds for

the efficiencies that can be reached for any possible process bywhich the IR and optical energy can be transformed. Theseupper bounds are in some way the equivalent of the theoreticalefficiency upper bounds for conventional solar cells. This isdone for five metals, including the most popular ones used in

plasmonics. It is evident that our techniques can also be usedfor other metals.

Note that this invited paper is totally based on [12], a paper

where much more details can be found.

II. RECEIVED POWERMost solar radiation is in the visible and the near-infrared

wavelength region (Figure 1). In order to form an alternative

to state-of-the-art solar cells, nano-antennas have to be

designed for this part of the spectrum. Since the materialproperties in these bands may vary a lot with frequency,

studies in the lower frequency bands, as already discussed in

literature [1], may be useful to build up necessary know-how,

978-88-907018-3-2/13 2013 IEEE

7th European Conference on Antennas and Propagation (EUCAP 2013) - Convened Sessions

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but do not necessarily offer the solutions for the concrete

problems at hand in the IR and optical frequency range.

Having the specific application of solar energy harvesting inmind, a new parameter is proposed to characterize the

antenna: the total harvesting efficiency. This parameter is

defined as

Figure 1. The solar spectrum. The major part of the energy is located in the

visible and the near-infrared band. The contribution to the total energy of thepart above 1500 nm is very small.

0

0

( , ) ( )

( , )

rad

totP T d

P T d

=

(2)

where is the wavelength, ( )rad is the radiationefficiency of the nano antenna as a function of the wavelength,

and Pis Plancks law for black body radiation

1

12),(

5

2

=

kT

hc

e

hcTP

(3)

where Tis the absolute temperature of the black body (in K), his Plancks constant (6.626*10

-34Js), c is the speed of light in

vacuum (3*108 m/s), and k is the Boltzmann constant

(1.38*10-23

J/K). In the case of solar energy harvesting, the

temperature Tis the temperature of the surface of the sun. The

values calculated with (2) have to be considered upper bounds

since practical solar radiation is filtered by the atmosphere.

However, they provide an excellent figure of merit for the

nano-antenna topologies investigated.

Figure 2. The dipole model studied. a) W and H are set equal to 40 nm and the

gap G is fixed at 10 nm, b) The dipole as t ransmitting antenna with a model

for the feeding structure located in the gap. c) The dipole as receiving antennaexcited by a plane wave.

This work reports on a systematic numerical study of theradiation efficiency of IR and optical nano dipoles for fivedifferent metals. No Drude model is used but the experimentalvalues of the material parameters are used directly. The solverused in this work is based on the solution of volumetric integralequations with the Method of Moments (MoM) [8], [9].Following the advice given by Vasylchenko [10], it was first

benchmarked against another solver and against measurements.Benchmarking of the simulation tool against measurementswas done as part of previous studies [7], [11].

III. RESULTSThe dipole topology studied in this paper is depicted in Fig.

2. Since nano-dipoles have to be fabricated on a supportinglayer, the effect of a glass substrate is included. Also for theglass substrate the measured permittivities are used in theanalysis. For the frequency range considered this permittivity isalmost constant and about 2.1. The efficiencies as a function offrequency (or free space wavelength) for different thicknessesof the substrate are plotted in Fig. 3. In practice substrates arevery thick compared to the wavelength. The total efficiencieson a half space of substrate material are 61.6 % for silver, 34.3% for gold, 50.3 % for aluminum, 29.5 % for copper, and 9.4% for chrome. This is a bit lower than the efficiencies invacuum. The results for vacuum can be found in [12]. Thislowering effect is caused by the interference of the field wavesreflected at the interfaces of the substrate. The majorconclusion of this study is that 1. the presence of a substrateoverall decreases the efficiencies, and 2. considering the samethickness of the glass, silver is by far superior over the othermaterials.

IV. EXTRACTION OF THE EFFECT OF THE MATERIALPROPERTIES

It is worthwhile to explain these results from a physical

point of view. Through (2), the total harvesting efficiency is

completely determined by the radiation efficiency, which is a

function of frequency / wavelength. The key issue is to try to

reach the highest efficiencies around 500 nm, where the solar

irradiance is the largest. This can be done by choosing the

proper dipole length. The reason is that this length is one of

the main factors that determines the response of the dipole.

However, the properties of the material also have a strong

effect on the response. That explains why a different optimal

length is found for different materials. Also, efficiency is

totally depending on losses, and losses are determined by theimaginary part of the permittivity.

It is possible to approximately assess any arbitrary material

with respect to its harvesting capabilities. For this, we need to

separate the effect of the plasmonic waves travelling along the

dipole, and consequently the complex interferences, from the

effect of the material. This is easily done for an elementary

dipole antenna in free space, i.e. a dipole with very short

length with respect to the wavelength. For such an antenna the

radiated power is proportional to the square of the dipole

moment


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400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Wavelength, nm

RadiationEfficiency

d=0nm : tot

=64.1%

d=20nm : tot

=49.7%

d=200nm:tot

=28.8%

d=infinite:tot

=61.6%

Figure 3. Efficiencies for a 250 nm silver dipole in terms of the thickness of

the glass substrate.

400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Wavelength, nm

RadiationEfficiency

d=0nm : tot

=34.8%

d=20nm : tot

=27.6%

d=200nm:tot

=14.7%

d=infinite:tot

=34.3%

Figure 4. Efficiencies for a gold 250 nm dipole in terms of the thickness of the

glass substrate.

400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Wavelength, nm

Rad

iationEfficiency

d=0nm : tot

=48.9%

d=20nm : tot

=37.2%

d=200nm:tot

=25.8%

d=infinite:tot

=50.3%

Figure 5. Efficiencies for an aluminum 250 nm dipole in terms of the

thickness of the glass substrate.

400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Wavelength, nm

Radia

tionEfficiency

d=0nm : tot

=29.3%

d=20nm : tot

=23.5%

d=200nm:tot

=12.5%

d=infinite:tot

=29.5%

Figure 6. Efficiencies for a copper 250 nm dipole in terms of the thickness of

the glass substrate.

400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Wavelength, nm

RadiationEfficiency

d=0nm : tot

=10.0%

d=20nm : tot

=6.8%

d=200nm:tot

=5.0%

d=infinite:tot

=9.4%

Figure 7. Efficiencies for a chromium 250 nm dipole in terms of the thickness

of the glass substrate.

2

0 0 0 *( ) ( )12

rad

V V

P dV dV

= J J (4)

where Vis the volume of the dipole, and J is the vector current

distribution flowing in it. Using the relation between the total

electric field and the (polarization and conduction) current in

the material, i.e. 0( )j = J E , the losses can be

expressed as

* *

0

*

2 2

0

1 1Re( ( ) ) Re( ( ) )

2 2 ( )

( )2 (( ) )

loss

V V

im

re im V

P dV dVj

dV

= =

=

+

JE J J

J J

(5)

so that the efficiency becomes

*

3 2 2 *

0 0 0 0

1

( )6

1(( ) ) ( ) ( )

rad

im V

re im

V V

dV

dV dV

=

+

J J

J J

(6)


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For a standard short dipole with cross section A and length

L carrying a standard linear rooftop type of current, the

integrals over the current density can be analyticallycalculated, yielding

3 2 2

0 0 0 0

,

2 2 3

, , 0

1

8 11

(( ) )

1

81

(( 1) )

rad

im

re im

im r

re r im r

V

k V

=

+

=

+

(7)

Note that this expression confirms the fact that the imaginarypart of the permittivity always has to be negative. Keeping the

electrical volume of the dipole constant and small, it is easily

seen that the efficiency becomes depending only on the

permittivity. From a material perspective, the key issue is to

get the ratio,

2 2

, ,(( 1) )

im r

re r im r

+

as low as possible. This is

reached for a very low imaginary part with respect to the real

part of the permittivity, or, when the real part approaches 1,for a very large imaginary part. Expression (7) is illustrated in

Fig. 8 for the five metals considered while the electrical

volume is kept constant at3

01k V = .

400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Wavelength, nm

rad

Ag

Au

Al

Cu

Cr

Figure 8. Small dipole efficiencies for different materials.

This means that a different dipole size at each frequency isused in order to remove the effect of the topology itself. The

strong dependency of the efficiency on the permittivity isclearly visible. The steep rise around 400 nm and 600 nm, for

Ag on the one hand and Au and Cu on the other hand, is

clearly also seen in the efficiency curves, but modulated there

by the plasmonic waves and resonance. Fig. 8 reveals whysilver outclasses the other materials, and why aluminum is a

good alternative. They keep a high efficiency in the whole

frequency band of importance. It is important to note that

expression (7) can be easily used to preliminary assess the

intrinsic harvesting capabilities of arbitrary metals in a first

approximate way. A high efficiency is needed in a band as

wide as possible around 500 nm. This is only the case for

silver and aluminum.

V. CONCLUSIONSUpper bounds are derived for the efficiencies by which

energy can be harvested from the sun using nano-antenna

technology. To this goal, the parameter total harvesting

efficiency is introduced. Dipoles on a glass substrate areconsidered. A simple approximating formula is derived to

assess the intrinsic harvesting capabilities of a material.

Several challenges remain. Silver is more susceptible to

oxidation, which can completely destroy its superiority.

VI. ACKNOWLEDGEMENTS The authors would also like to acknowledge the financial

support from the Fund for Scientific Research-Flanders

(FWO-V), through the FWO project G.0897.10N, the

Katholieke Universiteit Leuven (GOA), and the Methusalem

Funding by the Flemish government.

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[2] [2] Robert F. Service, Solar cells gear up to go somewhere under therainbow, Science, 20 June 2008, pp. 1585.

[3] [3] H. Gao, K. Li, F. Kong, H. Xie, and J. Zhao, Optimizing nano-optical antenna for the enhancement of spontaneous emission, ProgressIn Electromagnetics Research, Vol. 104, 313-331, 2010.

[4] [4] J.-S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, ImpedanceMatching and Emission Properties of Nanoantennas in an OpticalNanocircuit, Nano Lett., Vol. 9, No. 5, pp 18971902, Sep. 2009.

[5] [5] M. W. Knight, H. Sobhani, P. Norlander, N. J. Halas, Photodetectionwith active optical antennas, Science, vol. 332, 6 May 2011.

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[7] G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov,On the use of the method of moments in plasmonic applications, RadioScience, VOL. 46, RS0E02, 6 PP., 2011 doi:10.1029/2010RS004582.

[8] M. Vrancken and G. A. E. Vandenbosch, "Semantics of dyadic andmixed potential field representation for 3D current distributions in planarstratified media," IEEE Trans. Antennas Propagat., Vol. 51, no. 10, pp.2778-2787, Oct. 2003.

[9] Y. Schols, G.A.E. Vandenbosch, Separation of horizontal and verticaldependencies in a surface/volume integral equation approach to modelquasi 3D structures in multilayered media, IEEE Transactions onAntennas and Propagation, vol. 55, no. 4, April 2007,pp. 1086-1094.

[10] A. Vasylchenko, Y. Schols, W. De Raedt, and G. A. E. Vandenbosch,Quality Assessment of Computational Techniques and Software Toolsfor Planar-Antenna Analysis, IEEE Antennas and PropagationMagazine, Vol. 51, No. 1, pp. 23-38, Feb. 2009.

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