Proceedings of the 7th Annual ISC Graduate Research Symposium
ISC-GRS 2013
April 24, 2013, Rolla, Missouri
METAMATERIAL ABSORBER/EMITTER BASED ON NANOWIRE CAVITIES FOR SOLAR
THERMO-PHOTOVOLTAIC SYSTEMS
Huixu Deng
Department of Mechanical and Aerospace Engineering,
Missouri University of Science and Technology,
Rolla, MO 65409, USA
ABSTRACT
According to Shockley-Queisser (SQ) limit, a narrowband
emission spectrum of the metamaterial absorber/emitter with the
emission peak located slightly above the bandgap of
photovoltaic (PV) cells is desired to improve the overall
efficiency of thermo-photovoltaic (TPV) systems. In this article,
a metamaterial absorber/emitter based on gold nanowire cavities
on a gold ground plane is designed to achieve such an emission
spectrum. This absorber/emitter made of gold nanowires
embedded in alumina host exhibits an effective permittivity with
strong anisotropy, which supports cavity resonant modes of both
electric dipole and magnetic dipole. The impedance of the
cavity modes matches the impedance in free space, leading to
the emission of photons on resonance with the desired energy
slightly above the bandgap. Simulation results show that the
designed metamaterial absorber/emitter is polarizationinsensitive and nearly omnidirectional for the light angle.
Theoretical calculation shows that the overall efficiency of the
TPV system can reach SQ limit at a low emitter temperature,
𝑇𝑒 = 1160 K, and even exceed it when the temperature is higher.
1. INTRODUCTION
Metamaterials with artificial structured composites can exhibit
intriguing electromagnetic phenomena, such as negative
refraction [1], invisible cloaking [2], near-zero permittivity [3],
and indefinite cavities [4, 5]. The macroscopic properties of
metamaterials can be tailored flexibly by designing the artificial
meta-atoms. Metamaterial absorber/emitter working at higher
frequencies such as THz and infrared range have been designed
and demonstrated [6-10], which hold great promise in light
harvesting, thermal detection and electromagnetic energy
conversion. In the application of solar energy harvesting using
PV cells, the main challenge is to improve the efficiency of TPV
systems in which the energy conversion process is from heat to
electricity via photons. The maximum efficiency is defined by
SQ limit, which arises mainly from two loss mechanisms: on
one hand, photons incident on the PV cells with energy lower
than the PV cell bandgap energy will not be absorbed by the PV
cell and lost in free space; on the other hand, photons with
energy above the bandgap energy can be absorbed, but the
excess energy will be lost in generating undesirable heat. One
way to avoid these two kinds of loss is to design a thermal
emitter whose resonance frequency locates slightly above the
PV cell bandgap energy with a narrow emission bandwidth.
Since the optical properties of metamaterial structures can
be easily tuned by changing the structure sizes, in this article, a
novel design of metamaterial absorber/emitter based on
nanowire metamaterial cavities is proposed, as shown in Fig.
1(b). Periodic nanowire metamaterial cavities with a period of
𝑃 = 600 nm are grounded by a 150 nm thick gold film on a
glass substrate. In each cavity, 6 by 6 gold nanowires with
radius 𝑟0 and center-to-center distance 𝐷 = 60 nm are
embedded in alumina host with a cube geometry of 𝐿 = 360 nm
and ℎ = 130 nm. Finite-element method (FEM) simulation is
performed to obtain the reflection spectrum 𝑅 . And the
absorption spectrum 𝛼 is 𝛼 = 1 − 𝑅, since the transmission is
exactly zero due to the thick gold ground plane. In addition,
according to Kirchhoff’s law of thermal radiation, the
absorption spectrum is equal to the emission spectrum, 𝜖 = 𝛼.
The permittivity of gold is described by Drude model 𝜀𝑚 (𝜔) =
1 − 𝜔𝑝2 /𝜔(𝜔 + 𝑖𝛾0 ) with plasma frequency 𝜔𝑝 = 1.37 ×
1016 rad/s and bulk collision frequency 𝛾0 = 4.08 ×
1013 rad/s. The permittivity of alumina is 𝜀𝑑 = 3.0625.
The absorption/emission spectrum of the nanowire
metamaterial cavities array with 𝑟0 = 12.5 nm at normal
incidence is shown as the blue curve in Fig. 1(c). The main
absorption/emission peak of almost 100% is achieved at the
frequency of 195.9 THz , together with a smaller peak at
143.5 THz . The normalized emission spectrum of ideal
blackbody 𝜖𝐵𝐵 is also presented as the black curve in Fig. 1(c) at
temperature of 𝑇 = 1160 K. And the thermal emissivity of the
metamaterial emitter 𝐸𝑒 at 𝑇 = 1160 K can be obtained by 𝐸𝑒 =
𝜖𝐵𝐵 × 𝜖 as the red curve in Fig. 1(c). According to the effective
medium theory (EMT), the gold nanowires embedded in
alumina host can be regarded as an anisotropic medium with the
following permittivity components [11],
𝜀𝑥 = 𝜀𝑦 = 𝜀𝑑
(1 + 𝑓𝑚 )𝜀𝑚 + (1 − 𝑓𝑚 )𝜀𝑑
,
(1 − 𝑓𝑚 )𝜀𝑚 + (1 + 𝑓𝑚 )𝜀𝑑
𝜀𝑧 = 𝑓𝑚 𝜀𝑚 + (1 − 𝑓𝑚 )𝜀𝑑
(1)
where 𝑓𝑚 = 𝜋𝑟02 /𝐷2 is the volume filling ratio of gold nanowire.
1
Figure 1. Schematic of the TPV system and the emission spectra of the metamaterial emitter. (a) is the TPV system model mainly
including three components: the heat source to provide thermal energy, metamaterial absorber/emitter to emit photons at the
specific frequency and the PV cell to collect photons and generate electricity. (b) is the structure of the metamaterial
absorber/emitter made of gold nanowires embedded in alumina host. (c) is the emission spectra in which the black curve is the
normalized emissivity of ideal blackbody 𝜖𝐵𝐵 at 𝑇 = 1160 K, the blue curve is the emissivity of the metamaterial absorber/emitter
𝜖 which is equal to its absorptivity according to Kirchhoff’s law, 𝜖 = 𝛼, and the red curve is the thermal emission spectrum of the
emitter, 𝐸𝑒 = 𝜖𝐵𝐵 × 𝜖 at 𝑇 = 1160 K. The PV cell can absorb photons with energy higher than the bandgap 𝐸𝑔 represented by the
yellow area.
2. ANALYSIS OF THE ABSORBER/EMITTER
The magnetic field profiles 𝐻𝑦 in the x-z plane of the cavity
resonant modes are shown in Fig. 2(c,d), where the 𝑚 = 1 mode
with one magnetic field peak along the x direction is located at
the main absorption/emission peak; while the 𝑚 = 3 mode
corresponds to the weak absorption/emission resonance . In the
x-y plane as shown in Fig. 2(a,b), the magnetic field profiles are
homogeneous along the y direction for both modes. The 𝑚 = 2
mode is absent at the normal incidence due to the structural
mirror symmetry (with respect to y-z plane) of the metamaterial
cavities. The mechanism of high emission for the 𝑚 = 1 mode
can be understood as the cavity mode supports the electric
dipole resonance and the magnetic dipole resonance
simultaneously. As shown in Fig. 2(e), the divergence and
convergence of electric field at the top left and top right corners
of the metamaterial cavity imply the accumulation of polarized
positive charges and negative charges (with ∇ ∙ 𝐸⃗ = 𝜌/𝜀0 ,
where 𝜌 is the polarized charge density), which serves as a
strong electric dipole. Fig. 2(f) shows the distribution of
displacement current 𝐷 , with 𝜕𝐷/𝜕𝑡 = −𝑖𝜔𝐷 = −𝑖𝜔𝜀𝐸 .
Strong displacement current is found inside the grounded gold
film due to the large negative permittivity of metal. Anti-parallel
displacement currents are formed between the nanowire cavity
and the gold ground plane, resulting in a strong magnetic dipole
resonance, which is also shown in Fig. 2(f). The presence of
both electric resonance and magnetic resonance can result in
matched impedance to the free space, i.e., high
absorption/emission of photons on resonance. The current
absorber/emitter works in a narrow bandwidth due to the
resonance nature of light absorption/emission.
To ensure the emission resonant frequency is slightly above
the bandgap energy for different PV cells, the geometries of the
absorber/emitter structures can be tuned accordingly. Since the
impedance matching condition is critical for obtaining high
absorption/emission, it is expected that 𝑓𝑚 will largely affect the
performance of the metamaterial
Figure 2. (a) and (b) are the magnetic field profiles of the m
= 1 and m = 3 resonant modes in the x-y plane, respectively;
while (c) and (d) are in the x-z plane. (e) is the intensity and
direction of the electric field and (d) is the displacement
current for 𝑚 = 1 mode at 𝑦 = 0.
absorber/emitter. Fig. 3(a,b) shows the dependence of
absorption/emission spectra on 𝑓𝑚 at incident angle 𝜃 = 0° with
different collision frequencies of gold 𝛾, where the variation of
filling ratio is derived by tuning the radius of the gold nanowires
from 𝑟0 = 10 nm to 𝑟0 = 20 nm. Considering that the collision
2
frequency 𝛾 increases a lot due to the inevitable surface
roughness of the fabricated sample, numerical simulations with
𝛾 = 𝛾0 and 𝛾 = 3𝛾0 are shown in Fig. 3(a) and Fig. 3(b),
respectively. For 𝛾 = 𝛾0 , the main absorption/emission peak is
obtained around 𝑓𝑚 = 0.14 . For 𝛾 = 3𝛾0 , the highest
absorption/emission occurs at a larger filling ratio around 𝑓𝑚 =
0.25. The absorption/emission spectra of 𝛾 = 3𝛾0 show much
broader bandwidth compared to those of 𝛾 = 𝛾0 . By comparing
the absorption/emission spectra of the two collision frequencies,
it is also found that the resonance for the 𝑚 = 3 mode becomes
much weaker at 𝛾 = 3𝛾0 since high order resonance is very
sensitive to the material loss. Besides, there is an additional
resonance peak between the 𝑚 = 1 and the 𝑚 = 3 modes at
large filling ratios (𝑓𝑚 > 0.22) when 𝛾 = 𝛾0 . This is a high
order mode oscillating along the y direction.
𝜃 increases, while the peak remains the same. The
absorption/emission of TM polarized light, on the other hand,
decreases in a slower way than that of TE polarized light, while
the peak slightly shifts with the growth of incident angle.
Nevertheless, the absorption/emission remains strong for
incident angle up to 80° for both polarizations. An additional
peak is noted for incident angle larger than 40° in Fig. 3(d) and
3(f), which is the 𝑚 = 2 mode excited by the oblique incident
light.
3. EFFICIENCY OF THE TPV SYSTEM
To evaluate the performance of the metamaterial emitter made
of nanowire cavities for TPV systems, the overall system
efficiency is theoretically calculated. In general, the maximum
efficiency of TPVs is limited by the Carnot efficiency (all heat
energy is converted to electrical energy without loss) which can
be given by 𝜂𝑐𝑎𝑟 = 1 − 𝑇𝑐 /𝑇𝑒 . When the temperature of the PV
cell collector is 300 K and the temperature of the emitter is
maintained at 1000 K , a maximum efficiency of 0.7 can be
obtained. However, for TPV systems, efficiencies are limited by
other factors in the energy conversion process from heat to
electricity via photons. Following the analysis by Shockley and
Queisser, the overall energy conversion efficiency for TPVs is
limited by [12]
𝜂 = 𝑈(𝑇, 𝐸𝑔 ) × 𝜈(𝑇, 𝐸𝑔 ) × 𝑚(𝑉𝑜𝑝 )
Figure 3. (a) and (b) are the dependence of absorption
spectra on metal filling ratio at normal incidence with
different collision frequencies of gold, 𝛾 = 𝛾0 and 𝛾 =
3𝛾0 , respectively. (c-f) are the dependence of light
absorption on the incident angle 𝜃 for different
polarizations and collision frequencies with 𝑓𝑚 = 0.14 in
(c, d) and 𝑓𝑚 = 0.25 in (e, f).
Since the metamaterial cavities array possesses 4-fold
rotation symmetry in the x-y plane, it is polarizationindependent at normal incidence. For oblique incidence,
however, the absorption/emission performance will depends on
the polarization. The absorption/emission spectra for both
polarizations (transverse electric (TE) polarization and
transverse magnetic (TM) polarization) at two different collision
frequencies are shown in Fig. 3(c-f). The absorption/emission of
TE polarized light will decrease gradually as the incident angle
(2)
where 𝑈 is the so-called ‘ultimate efficiency’ which is
proportional to the energy contained in photon-excited electronhole pairs divided by the incident radiation energy, 𝜈 is an
efficiency due to recombination process which means charge
carries may be removed from the PV cell in additional ways,
and finally 𝑚 is called ‘impedance mismatch’ efficiency caused
by the difference between the open circuit voltage 𝑉𝑜𝑝 and the
optimal operating voltage.
According to the assumption that one photon with energy
larger than the PV cell energy gap 𝐸𝑔 will excite one electronhole pair resulting in a contribution of 𝐸𝑔 electricity energy to
the TPV system, the ultimate efficiency can be achieved by
calculating the energy carried by excited electron-hole pairs
with respect to the incident radiation energy [12, 13]:
𝜋/2
𝑈=
∫0
∞
𝑑𝜃 sin(2𝜃) ∫𝐸 𝑑𝐸𝜖(𝐸, 𝜃)𝐼𝐵𝐵 (𝐸, 𝑇𝑒 )
𝑔
𝜋/2
∫0
∞
𝐸𝑔
𝐸
𝑑𝜃 sin(2𝜃) ∫0 𝑑𝐸𝜖(𝐸, 𝜃)𝐼𝐵𝐵 (𝐸, 𝑇𝑒 )
(3)
In this equation, the denominator is the incident radiation
energy onto the PV cell, which is equal to the thermal emission
energy from the metamaterial emitter calculated by the integral
of blackbody radiation 𝐼𝐵𝐵 (when the emitter temperature is 𝑇𝑒 )
multiplying the emission of the emitter at all angles 𝜃 (assuming
the emitter to be planar with no azimuthal angular dependence).
The numerator is the energy contained in excited electron-hole
pairs by cutting off the photon energy below 𝐸𝑔 and reducing
the higher photon energy into 𝐸𝑔 . In order to average the
3
Figure 4. Theoretical calculation of the TPV system efficiency as a function of the PV cell bandgap energy 𝐸𝑔 and the emitter
temperature 𝑇𝑒 . (a) is the ultimate efficiency 𝑈, which is optimized when either of the two resonance peaks (195.9 THz → 0.8 eV
and 143.5 THz → 0.59 eV) of the emission spectrum is slightly above the bandgap 𝐸𝑔 . The dashed line represents the bandgap
energy of GaSb, 𝐸𝑔 = 0.71 eV. (b) and (c) are the recombination efficiency 𝜈 and the impedance mismatch efficiency 𝑚 ,
respectively, which will decrease when the emitter temperature is decreased. (d) is the overall conversion efficiency, 𝜂 = 𝑈 × 𝜈 ×
𝑚, which can reach the SQ limit of 0.31 when the PV cell is GaSb (represented by the horizontal dashed line, 𝐸𝑔 = 0.71 eV) and
the temperature is relatively low (represented by the vertical dashed line, 𝑇𝑒 = 1160 K).
emission in both TE and TM polarizations, we take 𝜖(𝐸, 𝜃) =
𝜖 𝑇𝐸 (𝐸, 𝜃)/2 + 𝜖 𝑇𝑀 (𝐸, 𝜃)/2.
Besides the efficiency of creating excited electron-hole
pairs, the recombination efficiency 𝜈 is utilized to measure the
percent of charge carriers becoming the desired external current.
And it can be expressed as the ratio between the open circuit
voltage 𝑉𝑜𝑝 and the initial material bandgap voltage 𝑉𝑔 = 𝐸𝑔 /𝑞,
in which 𝑞 is the charge of one electron [13, 14]:
𝑉𝑜𝑝 𝑉𝑐
𝑄𝑒 (𝑇, 𝐸𝑔 )
𝜈=
= ln (𝑓
)
𝑉𝑔
𝑉𝑔
𝑄𝑐 (𝑇𝑐 , 𝐸𝑔 )
(4)
where 𝑉𝑐 = 𝑘𝐵 𝑇𝑐 /𝑞 is the voltage of the PV cell related to the
cell temperature 𝑇𝑐 . 𝑄𝑒 and 𝑄𝑐 are photon number flux incident
on the PV cell from the emitter, and from an ideal blackbody
surrounding the cell at the same temperature of the PV cell 𝑇𝑐 ,
respectively, as shown in Eq. (5) [13]. Here, 𝑓 is chosen as 0.5
for an ideal case when both the emitter and the cell have planar
geometries.
𝜋
∞
2𝜋 2
𝐸 2 𝑑𝐸
𝑄𝑒 (𝑇, 𝐸𝑔 ) = 3 2 ∫ sin(2𝜃) 𝑑𝜃 ∫ 𝜖𝑒 (𝐸, 𝜃)
,
𝐸
ℎ 𝑐 0
𝐸𝑔
exp (
)−1
𝑘𝐵 𝑇
2𝜋 ∞
𝐸 2 𝑑𝐸
𝑄𝑐 (𝑇𝑐 , 𝐸𝑔 ) = 3 2 ∫
(5)
ℎ 𝑐 𝐸𝑔 exp ( 𝐸 ) − 1
𝑘𝐵 𝑇𝑐
The third factor in Eq. (2), 𝑚(𝑉𝑜𝑝 ), is evaluated when the
operating voltage 𝑉𝑚𝑎𝑥 is chosen to maximize the electrical
power for the PV cell and given by Ref. [13]:
𝑚(𝑉𝑜𝑝 ) =
2
𝑧𝑚
(1 + 𝑧𝑚 − 𝑒 −𝑧𝑚 )(𝑧𝑚 + ln(1 + 𝑧𝑚 ))
(6)
with 𝑧𝑚 determined by the ratio between 𝑉𝑚𝑎𝑥 and 𝑉𝐶 , and
related to
𝑧𝑜𝑝 =
4
𝑉𝑜𝑝
= 𝑧𝑚 + ln(1 + 𝑧𝑚 )
𝑉𝑐
(7)
When the filling ratio of nanowire in the emitter is 𝑓𝑚 =
0.14 and the collision frequency of gold is 𝛾 = 𝛾0 as shown in
Fig. 3(c,d), each efficiency factor in Eq. (2), 𝑈, 𝜈 and 𝑚, is
calculated and plotted in Fig. 4, as functions of the PV cell
bandgap energy 𝐸𝑔 and the temperature of the emitter 𝑇𝑒 . While
𝜈 and 𝑚 are both determined by the PV cell properties, 𝑈 is
enhanced by the designed metamaterial emitter so that the
overall efficiency of the TPV system is improved. From Fig.
4(d), the maximum efficiency locates at 𝐸𝑔 = 0.7~0.8 eV ,
where the resonance of the emitter (195.9 THz corresponding to
0.8 eV) is slightly above the bandgap energy 𝐸𝑔 . As mentioned
above, the explanation is that photons with energy below 𝐸𝑔
cannot be absorbed by the PV cell, however, for the photons
with energy above 𝐸𝑔 , the additional energy ∆𝐸 = 𝐸 − 𝐸𝑔 is
lost to thermalization within the PV cell. Here, the resonance of
the emitter is tuned at 195.9 THz by changing the filling ratio to
well match the gallium antimonide (GaSb) PV cell with a
bandgap energy of 0.71 eV . In this case, the needed emitter
temperature is about 𝑇𝑒 ≈ 1160 K to reach the SQ limit, 𝜂𝑆𝑄 =
0.31. This temperature is much lower than that in practical solar
TPV systems which is less than 2500 K. If the temperature is
too low, the efficiency will decrease due to the decrease of 𝜈
and 𝑚. When the temperature increases, the efficiency can even
get higher and exceed the SQ limit, but it is limited by the
melting point of the emitter materials [15]. The melting
temperature of gold is around 1337 K, which is much lower than
the current operation temperature of 1160 K. Therefore, new
designs of the emitter with high melting point materials may be
possible to improve the overall efficiency furthermore at high
temperature.
4. CONCLUSIONS
In conclusion, in order to improve the overall efficiency of TPV
system, a metamaterial absorber/emitter based on nanowire
cavities is proposed and demonstrated. The cavity resonant
mode with mode order 𝑚 = 1 is utilized to excite both the
electric dipole and the magnetic dipole simultaneously. At a
specific frequency, the impedance of the metamaterial
absorber/emitter can match to the free space, and the strong
optical loss in nanowire structures results in high emissivity
with narrow emission bandwidth. The resonant frequency can be
tuned to match the bandgap energy of different PV cells by
changing the filling ratio of gold nanwires. The
absorption/emission depending on different incident angles for
both the TE and TM polarizations is also investigated and
utilized in calculating the efficiency of TPV systems. The
overall conversion efficiency consists of three parts: the ultimate
efficiency 𝑈, the recombination efficiency 𝜈 and the impedance
mismatch efficiency 𝑚 . By tuning the emission resonant
frequency slightly above the PV cell bandgap energy, the
overall efficiency can reach SQ limit 𝜂𝑆𝑄 = 0.31 at a low
temperature of 1160 K. When the temperature of the emitter is
increased, the efficiency can exceed the SQ limit, but it is still
limited by the melting point of the emitter materials.
5. ACKNOWLEDGEMENTS
This work was partially supported by the Department of
Mechanical and Aerospace Engineering, and the Intelligent
Systems Center at Missouri S&T. The authors thank Y. He and J.
Gao for helpful discussions.
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