- Email: [email protected]
Meshed doped silicon photonic crystals for manipulating near-field thermal radiation
Accepted Manuscript
Meshed Doped Silicon Photonic Crystals for Manipulating Near-Field Thermal Radiation Mahmoud Elzouka , Sidy Ndao PII: DOI: Reference:
S0022-4073(17)30357-6 10.1016/j.jqsrt.2017.09.002 JQSRT 5828
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
10 May 2017 5 September 2017 6 September 2017
Please cite this article as: Mahmoud Elzouka , Sidy Ndao , Meshed Doped Silicon Photonic Crystals for Manipulating Near-Field Thermal Radiation, Journal of Quantitative Spectroscopy & Radiative Transfer (2017), doi: 10.1016/j.jqsrt.2017.09.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights
Significant enhancement in near-field thermal radiation is realized using practical and achievable micrometric gap, rather than nanometric gaps considered in majority of literature.
Near-field thermal radiation between meshed photonic crystal structure is investigated, for
CR IP T
the first time in literature. Enhancement in radiative heat transfer for meshed structures is 22 times higher than nonmeshed ones, which is promising for thermal rectification and thermal management applications.
First-principle method is used to simulate the random thermal vibrations of charges within
AN US
the material (i.e., the main mechanism of electromagnetic thermal emission).
Resonant modes validated and visualized by solving Helmholtz equation using weak-form
AC
CE
PT
ED
M
formulations.
Page 1 of 25
ACCEPTED MANUSCRIPT
Meshed Doped Silicon Photonic Crystals for Manipulating NearField Thermal Radiation Sidy Ndao, corresponding author University of Nebraska-Lincoln Mechanical and Materials Engineering Lincoln, Nebraska, 68588-0526, USA [email protected]
AC
CE
PT
ED
M
AN US
CR IP T
Mahmoud Elzouka University of Nebraska-Lincoln Mechanical and Materials Engineering Lincoln, Nebraska, 68588-0526, USA [email protected]
Page 2 of 25
ACCEPTED MANUSCRIPT
Meshed Doped Silicon Photonic Crystals for Manipulating NearField Thermal Radiation
CR IP T
Abstract The ability to control and manipulate heat flow is of great interest to thermal management and thermal logic and memory devices. Particularly, near-field thermal radiation presents a unique opportunity to enhance heat transfer while being able to tailor its characteristics (e.g., spectral
AN US
selectivity). However, achieving nanometric gaps, necessary for near-field, has been and remains a formidable challenge. Here, we demonstrate significant enhancement of the near-field heat transfer through meshed photonic crystals with separation gaps above 0.5 μm. Using a firstprinciple method, we investigate the meshed photonic structures numerically via finite-difference
M
time-domain technique (FDTD) along with the Langevin approach. Results for doped-silicon meshed structures show significant enhancement in heat transfer; 26 times over the non-meshed
ED
corrugated structures. This is especially important for thermal management and thermal
PT
rectification applications. The results also support the premise that thermal radiation at micro scale is a bulk (rather than a surface) phenomenon; the increase in heat transfer between two
CE
meshed-corrugated surfaces compared to the flat surface (8.2) wasn’t proportional to the increase in the surface area due to the corrugations (9). Results were further validated through good
AC
agreements between the resonant modes predicted from the dispersion relation (calculated using a finite-element method), and transmission factors (calculated from FDTD).
Page 3 of 25
ACCEPTED MANUSCRIPT
1. Keywords Near-field thermal radiation; photonic crystals; finite-difference time-domain; first-principle; cluster computing; meshed structures..
CR IP T
2. Introduction Manipulating heat transfer is of engineering interest for its potential applications in thermal storage, thermal management and thermal logic [1] and memory [2–4] devices. However,
AN US
controlling heat transfer is a challenge due to the lack of perfect thermal insulators, unlike in the case of electricity. Vacuum is an efficient thermal insulator, however not perfect since radiative heat transfer can still flow through vacuum. Nevertheless, using near-field effects, thermal radiation enhancement and/or suppression through vacuum can result in desired tailoring of heat
M
transfer.
ED
It has been demonstrated that near-field thermal radiation can enhance the radiative heat transfer dramatically, even beyond the classical theoretical limit of the blackbody radiation [5,6]. The
PT
transfer of thermal energy by near-field radiation occurs when a thermal emitter and receiver are brought very close to each other, at distances near or below the characteristic wavelength of
CE
thermal radiation. At this close proximity, in addition to propagating harmonic electromagnetic waves (as in the case of blackbody radiation), evanescent (i.e., non-propagating) waves which
AC
are confined to the thermal emitter’s surface (i.e., can only propagate along the surface) participate in the thermal energy transfer. The extra channels for heat flow created by these evanescent waves increase the rate of heat transfer exchange exponentially with decreasing separation gap (i.e., the smaller the gap, the stronger the evanescent fields are and the higher the thermal radiation exchange rate) [7,8]. These particularities of near-field thermal radiation make Page 4 of 25
ACCEPTED MANUSCRIPT
it ideal for thermal modulation applications, such as with thermal memory, thermal diode [9–13] and thermal switches [14,15]. Near-field thermal radiation has been of interest to scientists since the 1960s [16,17], and it has become increasingly an engineering research topic for the past two decades with advances in
CR IP T
nano/microfabrication techniques. Research efforts to enhance near-field thermal radiation range from the use of nano/microstructures [18–23], thin layers of polar dielectrics [24,25], patterned layers of graphene [26,27], extremely-thin photonic crystals [28] and metal-dielectric structures
AN US
that have hyperbolic dispersion [29–31]. Despite the numerous approaches, very few are
practical from the engineering standpoint; thin films of polar dielectrics [32] would be an example. Most significant engineering barriers include the fabrication of nanometric separation gaps and the implementation of exotic materials such as graphene.
M
In this paper, we are introducing a practical method for enhancing near-field thermal radiation,
ED
without necessarily using extremely small nanometer gaps. We propose using meshed photonic crystals with variable separation gaps as shown in Figure 1. The meshing increases the area
PT
available for heat transfer by thermal radiation and introduce new resonant electromagnetic modes. Both effects are expected to enhance the rate of radiative heat transfer. The meshed
CE
photonic crystals have a minimum separation gap of 0.5 μm, achievable with standard microfabrication techniques (i.e., projection lithography and deep reactive ion etching). We used
AC
gratings since they are one of the simplest structures and have commonly been used in optics applications. The photonic crystals are made out of doped-silicon which is fully compatible with most standard microfabrication techniques, well characterized, and most importantly, its optical properties can be controlled by varying its doping level [33]; it can be engineered to act as an opaque metal or transparent dielectric. We investigated the meshed photonic crystal numerically Page 5 of 25
ACCEPTED MANUSCRIPT
using a first principle finite-difference time-domain technique (FDTD). The technique solves Maxwell’s equations with added fluctuating current source (to simulate thermal radiation source). We investigate the effect of tooth depth and meshing on the heat transfer and discuss the resonant modes leading to the observed enhancement in heat transfer. In parallel, we calculated
CR IP T
the dispersion relation using finite-element method to verify the resonance frequencies in heat transfer from the FDTD results.
3. Methods
defined by its period , tooth height , and base thickness
AN US
The geometry of the meshed photonic crystal investigated in this study is shown in Figure 1. It is , spacing between meshed teeth
, separation distance
. We model the near-field thermal radiation using a first-principle
technique that simulate the source, scattering and absorption of thermal electromagnetic waves.
M
We use finite-difference time-domain (FDTD) method to solve Maxwell’s equations with
ED
random fluctuating current density source to represent the origin of thermal radiation. We adopted the same method introduced by Luo et. al. [34] to simulate the fluctuating current based
PT
on the Langevin approach. With this method, a simple variation is made to the FDTD algorithm [35] by introducing a randomly fluctuating component (in both magnitude and direction) to the
CE
polarization equation. The polarization equation defines the material’s polarization response due
AC
to local electric field, and it is used to update the value of the electric field in time stepping through the FDTD algorithm [34]
(1)
Page 6 of 25
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
is the vacuum permittivity. ,
CE
where
PT
Figure 1: Meshed photonic crystal layout (a) and computational domain (b-c) and
are the parameters of the Lorentz damped-
harmonic oscillator model that describes the polarization response to electric field.
AC
frictional coefficient or scattering rate, resonant frequency.
is the plasma frequency, and
is the
is the polarization
is a random variable added to the polarization equation to introduce
thermal fluctuations. In this work, we use doped-silicon which optical properties in the infrared regime (λ > 2 μm) can be modeled using the Drude model [33]. The photonic crystal structures
Page 7 of 25
ACCEPTED MANUSCRIPT
are modeled using heavily doped-silicon (p-type model parameters:
) with the following Drude
,
, and
.
To simulate the source of thermal radiation, the random fluctuations in the current density need
〈
〉
Where
(
)
is the current density in direction
(x, y, or z),
(2)
is the mean energy of Planck’s
AN US
oscillator;
CR IP T
to follow the fluctuation-dissipation theorem:
are Dirac delta functions, indicating that currents are uncorrelated in
M
and
(3)
both spatial space and frequency domain.
is the Kronecker delta which equal 1 for
ED
and zero otherwise, for an isotropic medium.
,
is the imaginary part of the relative permittivity
PT
of the material incorporating the fluctuating current, and
is the permittivity of free space.
To achieve fluctuations in current density that satisfies equation (2),
needs to have a
〉
(4)
AC
〈
CE
frequency profile of the form [34]:
where
is a constant comprising material’s parameters and volume of discretization elements
used in FDTD. The correlation in equation (4) is colored noise and it is possible to incorporate it in the FDTD algorithm. However, there are two inconveniences: the simulation will be only valid for a fixed temperature (used to calculate
Page 8 of 25
), and the generating colored noise needs
ACCEPTED MANUSCRIPT
more computation power and memory than the simple Gaussian white-noise [36]. Therefore here, we exploit the linearity of Maxwell’s equations and assume fluctuations in
as a white-
noise with flat spectral density (i.e., constant amplitude in frequency domain). The resulted solutions are scaled with respect to
to account for the temperature-dependent current
CR IP T
fluctuations.
Since the simulated structures have a discrete periodicity in x-direction and continuous
periodicity in z-direction, we simulate only one unit cell of the structure (as shown in Figure 1),
AN US
and infer the solution of the infinitely periodic structure using Bloch’s theorem [37]. Bloch’s theorem dictates Bloch periodic boundary condition at the boundaries normal to directions of periodicity (i.e., x and z directions), which describes the relationship between the fields (e.g.,
is the period or the length of the unit cell in x-direction, and
ED
where
M
electric field ) at the two adjacent sides as;
(5) is the Bloch
wavevector that dictates the phase shift between the two sides of the unit cell. The solution of the
PT
electromagnetic problem of the infinitely periodic structure in x direction is the superposition of
CE
all the solutions of a unit cell with Bloch periodic boundary condition, with all the values of that satisfy unique solutions (note that
of
gives the same solution). The range
which satisfies unique solutions is termed as Brillouin zone [37]. We can choose any range
AC
of
and
on condition that we have a span of
, so we have chosen
range to be
. Since our unit cell is symmetric about x axis, we expect that solutions with
identical to solutions with
. Therefore, we can solve our problem only for the range
Page 9 of 25
to be
ACCEPTED MANUSCRIPT
. Here, we assumed a constant value of values of
, and we didn’t solve for different
since they have a minimal effect on the spectral shape of the flux [18,28].
The boundaries of the unit cell normal to the y-direction are open boundaries with no reflection
CR IP T
to electromagnetic waves. To simulate an open boundary, we use an artificial perfectly matched layer (PML) [38] with thickness
and padding distance
, see Figure 1.
To calculate the net thermal radiative energy transfer from the emitter to the receiver, or the energy absorbed by the receiver, we calculate the difference between electromagnetic energy
AN US
crossing the two boundaries denoted by the dashed lines in Figure 1 b-d. The energy flux can be calculated by integrating the Poynting vector over the surface. The energy flux absorbed by the receiver (the bottom structure), due to randomly-fluctuating current distributed over the emitter (the upper structure) at a particular Bloch wavevector is denoted by
M
fluctuations in the simulation are generated from a white-noise
. Since the current
term, the actual heat flux
ED
corresponding to a certain Bloch wavevector should be represented by total heat flux is obtained by integrating over the
CE
∫
spanning over one Brillouin zone (i.e.,
PT
);
, and the
(6)
AC
In this study, we run the simulation with
step of
equal to the surface emissivity (or absorptivity) at far-field at ), and we named
. We scaled the
to be
(i.e., for a blackbody,
the spectral transmission factor. Since we incorporate
random values in our simulation, the results include noise which can be cancelled out by repeating the simulation over a number of times (~ 30) and averaging the results from all the Page 10 of 25
ACCEPTED MANUSCRIPT
runs. The number of averaged simulation runs depends on the running time for each individual run. In this work, we utilized a free open-source implementation of the FDTD algorithm called Meep (an acronym for MIT Electromagnetic Equation Propagation) [35]. We used finite-element method to investigate the dispersion relation of resonant modes created
CR IP T
between the toothed structures and to compare them to resonant frequencies and wavevectors from FDTD simulations. The method is based on converting the Helmholtz eigenvalue equation to a weak form, which can be solved using any finite-element technique. The details are
AN US
explained by Parisi et. al. [39]. This method incorporates the material dispersion and losses,
AC
CE
PT
ED
M
which resulted in accurate prediction of resonant frequencies.
Page 11 of 25
ACCEPTED MANUSCRIPT
4. Results and discussion 4.1. Effect of tooth height We investigate the effect of tooth height on the near-field thermal radiation. For transverse
and two photonic crystals designs with tooth heights of
and
for flat slabs
CR IP T
magnetic (TM) waves, Figure 2 shows TM waves transmission factors
and tooth spacing
. The transmission factor is calculated between two structures with spacing
.
For the sake of validation, we have plotted the transmission factor calculated analytically using
wave vector
can be estimated by ∑
AN US
Green’s function method [40] for the flat slab case; transmission factor corresponding to a Bloch , where
is an integer. Resonance
peaks appear on the corrugated structures for all values of the Bloch wavevector as seen on
M
Figure 2, unlike with the flat slab. The corrugated structures add new resonant modes which act as additional channels to tunnel photons from the emitter to the receiver. The -
-height
ED
toothed structure results in higher number of resonant modes in comparison to the -
-height
one at lower frequencies. This can be explained by taking a closer look at the magnetic field of
PT
resonant modes that is shown in Figure 2-b. The first three magnetic field profiles (A1, A2 and and -
-height
CE
A3) and the second three profiles (B1, B2 and B3) are corresponding to
toothed structures, respectively. The magnetic field profiles show waveguide-like modes of first,
AC
second and third orders, which can be distinguished by counting the number of nodes inside the cavity. Note that the fields don’t look uniform due to the losses incorporated in the prediction of the field profiles, an advantage of using the weak form formulation in solving the Helmholtz eigenvalue equation [39]. The waveguide-like modes are known to have a lower cut-off frequency (i.e., the frequency of the lowest order mode) for a larger cavity height. Therefore, the
Page 12 of 25
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
PT
Figure 2: Effect of tooth height on the transmission factor for TM waves
-height toothed structure shows resonant modes at lower frequencies than the -
AC
-
CE
a) spectral transmission factor for different tooth height at different Bloch wavevectors. Transmission factors for the flat slab are calculated analytically and represented by dotted lines. b) Plot of the magnetic field component in z direction for resonant modes at locations marked in (a) and marked in the dispersion relation in Figure 4-b. Mean energy of Planck’s oscillator is scaled to fit the plot. -deep
ones. It is also important to note from the plots the high transmission factors at infrared frequencies for the toothed structures at high values of Bloch wavevectors ( above) in comparison to the nearly zero transmission factor for the flat surface.
Page 13 of 25
and
ACCEPTED MANUSCRIPT
The additional resonant modes introduced by the corrugated structures and their prevalence at large Bloch wavevectors result in enhancement of the radiative heat transfer over the flat slab as shown in the integrated transmission factor
at the bottom of Figure 2-a and explicitly
supported by the total heat transfer plots in Figure 5-b. However, the enhancement in
CR IP T
transmission factors from these resonant modes applies only to TM waves. For transverse electric (TE) waves, there are no resonant modes since the doped-silicon has no magnetic
response, and the photon tunneling due to total internal reflection is dominant. The larger the surface area through which the frustrated total internal reflection occurs, the higher the expected
AN US
heat transfer. Therefore, for TE waves, the flat slab shows higher rate of heat transfer than the corrugated structures separated by
gap, as shown in Figure 5-a. This is because the flat
slabs have higher heat transfer surface area at small gaps than the corrugated surfaces. The case
ED
4.2. Effect of tooth meshing
M
is different when meshing occurs, as will be discussed in the next section.
It is intuitive from the experience with heat transfer at macro scale to expect the rate of heat
PT
transfer to be proportional to the surface area. This picture is valid as long as the heat transfer is a surface phenomenon (i.e., thermal radiation is emitted and absorbed by a small layer close to the
CE
surface). The case is different at microscale, where we need to consider the thermal radiation
AC
absorbed by and emitted from points across the entire material (i.e., bulk), such as in the case of the present meshed photonic structures. Figure 3-a shows the transmission factors for two corrugated structures with tooth height of 8 μm as a function of separation gap, dc, as shown in the schematic in Figure 3-b. Negative values of dc correspond to cases where meshing occurs. Figure 3-a shows a decrease in the number of resonant modes at low
values with decreasing
separation gap. This is due to the reduction in the volume available for the resonant waveguidePage 14 of 25
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 3: Effect of structures meshing on transmission factor.
CE
PT
(a) spectral transmission factor for different separation gaps at different Bloch wavevectors for TM waves. (b) schematic shows the toothed structures at four different separation gaps of -7.5, -2.5, 0.5 and 10 . (c) magnetic field profiles for the first three modes of fully meshed structures with . The eigenmodes locations are indicated in (a) and in Figure 4-c. like modes. However, the smaller the separation gap, the higher the transmission factor of the
AC
resonant modes at higher
values, and the stronger the resonant modes are; note the absence of
resonant modes for structures with a separation distance of
at
separation gap results in enhancement of the integrated transmission factor the bottom plot of Figure 3-a.
Page 15 of 25
. Decreasing the , as shown at
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 4: Dispersion relation for resonant modes between two toothed structures
ED
M
All structures have the same period and tooth separation . a) dispersion relation for , and . b) and c) are dispersion relations for , and and , respectively. The grayscale intensity of each point represents the ratio in field intensity across one unit cell. The filled contour plot in (c) represents the spectral transmission factor, and it shows the agreement between the resonance points and the peaks in the transmission factor. Here, the most important effect of meshing is the enhancement in transmission factors from non-
PT
resonant modes (i.e., total internal reflection) in comparison to the non-meshed cases; this
CE
enhancement is clearly observable at frequencies away from the resonance frequencies. This contribution from non-resonant modes increases with decreasing separation gap. As can be seen
AC
on the integrated transmission factors plot, the fully meshed photonic crystal ( has almost no resonance, while the resonance is more noticed for the case of
) .
Dispersion relation for the fully-meshed structures (Figure 4-c) shows a continuum of states without any gap for all frequencies of interest. The dispersion relation is similar to case of wave propagating in isotropic medium, or a layered medium with low contrast in refractive index. This Page 16 of 25
CR IP T
ACCEPTED MANUSCRIPT
AN US
Figure 5: Total heat transfer scaled to heat transfer of flat slabs separated by 7-μm gap, for a) TE, b) TM and c)TE+TM contributions. case is known to have no band gaps and can support waves with all frequencies. The case is different from the wave guide-like dispersion shown in Figure 4-b for two separated structure,
M
which has a couple of pseudo band gaps.
ED
4.3. Effect of separation distance on total heat transfer We calculate the total heat transfer
∫
between the toothed structures for different
PT
designs and multiple separation distances; results are shown in Figure 5 for TE, TM and TE+TM waves. In the figure, the heat transfer
aparts. Since there are no TE surface resonant modes for non-magnetic material
CE
surfaces
is scaled to the total heat transfer between two flat
AC
structures [41], such being the case here, the radiative heat transfer for TE modes is mediated by propagating modes or evanescent modes from frustrated total internal reflection. Therefore, the radiative heat transfer is dependent on surface area. This can be seen on Figure 5-a at the separation gap of
, where the flat surfaces outperform corrugated structures due to the
higher surface area available for frustrated total internal reflection to occur. Note also the similar heat transfer values for the two structures, tooth height of Page 17 of 25
and
, but same
at
ACCEPTED MANUSCRIPT
the separation gap of
. This similarity is the result of the two surfaces having the same
overlapped surface area. The heat transfer rates for structures with the ones with
are higher than
, further supporting that heat transfer from TE modes depend heavily
surfaces [30].
CR IP T
on the surface area available for total internal reflection and the separation between those
With TM modes (Figure 5-b) however; the structured surfaces outperform the flat surfaces at all separation gaps, by over an order of magnitude for the case of
and
of
and
but different
AN US
Comparing the heat transfer for the two structures having the same
.
, we find that the values of heat transfer are almost identical when the
structures are meshed, although the separation gap between the two surfaces is different. This shows that the effect of resonant modes on the heat transfer is much stronger than the effects of
M
frustrated total internal reflection.
and
ED
It is worth noting that the heat transfer from TE (TM) modes at the fully meshed case for is higher than the case of heat transfer between two flat surfaces by a factor of
. The enhancement in heat transfer is
PT
with separation gap
expected to be enhanced by a factor equal to the increase in the surface area resulted from
CE
corrugation (
), based on the classical view of thermal radiation (i.e., radiative
AC
heat flux is proportional to the area of surfaces exchanging the heat). However, the enhancement in heat transfer was
for TE (TM) modes, which is less than the augmented surface area
ratio of 9. This has two reasons; the heat flux per unit surface area between parallel infinite surfaces should be higher than the heat flux between surfaces with a finite length of the structure’s teeth). The second reason is that tooth width,
, is thinner than the
penetration depth of radiation. Overall, fully meshed structures achieved enhancement in Page 18 of 25
(i.e.,
ACCEPTED MANUSCRIPT
radiative heat transfer 26 times higher than the non-meshed structures separated This enhancement could result in thermal rectifications (
apart.
) as high as 2500% making
meshed photonic crystals ideal for thermal management applications such as thermal memory
5. Conclusions
CR IP T
and thermal diodes.
We have investigated meshed photonic crystals to enhance near-field radiative heat transfer between two structures at a minimum separation gap of
. The goal is to achieve
AN US
enhancement in radiative heat transfer at practical separation gaps other than the nanometric gaps suggested in the literature. We have solved for the radiative heat transfer using a first-principle method; FDTD with the Langevin approach to simulate the thermal fluctuating currents. The
M
enhancement in the heat transfer from non-meshed corrugated structures relative to a flat slab has been mainly attributed to TM resonant modes. In the case of meshed photonic crystals, further
ED
enhancement in heat transfer resulted from the contribution of non-resonant modes (i.e., frustrated total internal reflection), which increases proportionally with surface area. We
PT
calculated the dispersion relation for the meshed and non-meshed cases using finite-element
CE
method; calculated resonant frequencies were in agreement with the results from the FDTD simulations. The dispersion relations for the non-meshed structures were similar to wave-guide
AC
resonant modes. In the case of meshed structures, the dispersion relation was similar to that of a homogenous media or layered media with low contrast in dielectric constant, which shows a continuum of states at all frequencies of interest to thermal radiation. Using meshed structures with a minimum separation gap of
, we were able to demonstrate enhancement in
radiative heat transfer as high as 26 times when compared to non-meshed structures. This
Page 19 of 25
ACCEPTED MANUSCRIPT
enhancement could result in thermal rectifications as high R=2500% (R = (Qmeshed – Qnon-meshed)/ Qnon-meshed ×100%); by incorporating the corrugated structures into NanoThermoMechanical rectifier [13], for instance. The high contrast in heat transfer between meshed and non-meshed cases makes meshed photonic crystals ideal for thermal management applications such as
CR IP T
thermal memory and thermal diodes.
Acknowledgment
This work was supported by the National Science Foundation (NSF) through the Nebraska
AN US
Materials Research Science and Engineering Center (MRSEC) (grant No. DMR-1420645). This work was completed utilizing the Holland Computing Center of the University of Nebraska, which receives support from the Nebraska Research Initiative. We would like to also thank Prof.
M
Christos Argyropoulos for the insightful conversations on electromagnetics modeling.
[1]
ED
References
N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, B. Li, Colloquium: Phononics: Manipulating
PT
heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84 (2012) 1045–1066.
[2]
CE
doi:10.1103/RevModPhys.84.1045. R. Xie, C.T. Bui, B. Varghese, Q. Zhang, C.H. Sow, B. Li, J.T.L. Thong, An electrically
AC
tuned solid-state thermal memory based on metal-insulator transition of single-crystalline VO2 nanobeams, Adv. Funct. Mater. 21 (2011) 1602–1607. doi:10.1002/adfm.201002436.
[3]
L. Wang, B. Li, Thermal memory: A storage of phononic information, Phys. Rev. Lett. 101 (2008). doi:10.1103/PhysRevLett.101.267203.
Page 20 of 25
ACCEPTED MANUSCRIPT
[4]
M. Elzouka, S. Ndao, Near-field NanoThermoMechanical memory, Appl. Phys. Lett. 105 (2014) 243510. doi:10.1063/1.4904828.
[5]
A. Narayanaswamy, S. Shen, L. Hu, X. Chen, G. Chen, Breakdown of the Planck
doi:10.1007/s00339-009-5203-5. [6]
CR IP T
blackbody radiation law at nanoscale gaps, Appl. Phys. A. 96 (2009) 357–362.
L. Hu, A. Narayanaswamy, X. Chen, G. Chen, Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law, Appl. Phys. Lett.
[7]
AN US
92 (2008) 133106. doi:10.1063/1.2905286.
B. Song, A. Fiorino, E. Meyhofer, P. Reddy, Near-field radiative thermal transport: From theory to experiment, AIP Adv. 5 (2015) 53503. doi:10.1063/1.4919048. R. St-Gelais, L. Zhu, S. Fan, M. Lipson, Near-field radiative heat transfer between parallel
M
[8]
structures in the deep subwavelength regime, Nat. Nanotechnol. (2016).
Z. Chen, C. Wong, S. Lubner, S. Yee, J. Miller, W. Jang, C. Hardin, A. Fong, J.E. Garay,
PT
[9]
ED
doi:10.1038/nnano.2016.20.
C. Dames, A photon thermal diode., Nat. Commun. 5 (2014) 5446.
CE
doi:10.1038/ncomms6446.
AC
[10] P. Ben-Abdallah, S.-A. Biehs, Phase-change radiative thermal diode, Appl. Phys. Lett. 103 (2013) 191907. doi:10.1063/1.4829618.
[11] C.Y. Tso, C.Y.H. Chao, Solid-state thermal diode with shape memory alloys, Int. J. Heat Mass Transf. 93 (2016) 605–611. doi:10.1016/j.ijheatmasstransfer.2015.10.045. [12] M.J. Martínez-Pérez, A. Fornieri, F. Giazotto, Rectification of electronic heat current by a Page 21 of 25
ACCEPTED MANUSCRIPT
hybrid thermal diode., Nat. Nanotechnol. 10 (2015) 303–7. doi:10.1038/nnano.2015.11. [13] M. Elzouka, S. Ndao, High Temperature Near-Field NanoThermoMechanical Rectification, Sci. Rep. 7 (2017) 44901. doi:10.1038/srep44901.
CR IP T
[14] W. Chung Lo, L. Wang, B. Li, Thermal Transistor: Heat Flux Switching and Modulating, J. Phys. Soc. Japan. 77 (2008) 54402. doi:10.1143/JPSJ.77.054402.
[15] W. Gu, G.-H. Tang, W.-Q. Tao, Thermal switch and thermal rectification enabled by nearfield radiative heat transfer between three slabs, Int. J. Heat Mass Transf. 82 (2015) 429–
AN US
434. doi:10.1016/j.ijheatmasstransfer.2014.11.058.
[16] E.G. Cravalho, C.L. Tien, R.P. Caren, Effect of Small Spacings on Radiative Transfer Between Two Dielectrics, J. Heat Transfer. 89 (1967) 351. doi:10.1115/1.3614396.
M
[17] S. Basu, Near-field radiative heat transfer across nanometer vacuum gaps: fundamentals and applications, 1st ed., Elsevier, Oxford, UK, 2016.
ED
http://www.sciencedirect.com/science/book/9780323429948 (accessed March 14, 2017).
PT
[18] J. Dai, S.A. Dyakov, M. Yan, Enhanced near-field radiative heat transfer between corrugated metal plates: Role of spoof surface plasmon polaritons, Phys. Rev. B. 92
CE
(2015) 35419. doi:10.1103/PhysRevB.92.035419.
AC
[19] X. Liu, B. Zhao, Z.M. Zhang, Enhanced near-field thermal radiation and reduced Casimir stiction between doped-Si gratings, Phys. Rev. A. 91 (2015) 62510. doi:10.1103/PhysRevA.91.062510.
[20] R. Guérout, J. Lussange, F.S.S. Rosa, J.-P.J.-P. Hugonin, D.A.R. Dalvit, J.-J.J.-J. Greffet, A. Lambrecht, S. Reynaud, Enhanced radiative heat transfer between nanostructured gold Page 22 of 25
ACCEPTED MANUSCRIPT
plates, Phys. Rev. B. 395 (2012) 180301. doi:10.1088/1742-6596/395/1/012154. [21] J. Lussange, R. Guérout, F.S.S. Rosa, J.-J. Greffet, A. Lambrecht, S. Reynaud, Radiative heat transfer between two dielectric nanogratings in the scattering approach, Phys. Rev. B.
CR IP T
86 (2012) 85432. doi:10.1103/PhysRevB.86.085432. [22] S.-A. Biehs, F.S.S. Rosa, P. Ben-Abdallah, Modulation of near-field heat transfer between two gratings, Appl. Phys. Lett. 98 (2011) 243102. doi:10.1063/1.3596707.
[23] A. Didari, M. Pinar Mengüç, Near-field thermal emission between corrugated Surfaces
doi:10.1016/j.jqsrt.2015.02.016.
AN US
separated by nano-Gaps, J. Quant. Spectrosc. Radiat. Transf. (2015).
[24] S.-A. Biehs, Thermal heat radiation, near-field energy density and near-field radiative heat
M
transfer of coated materials, Eur. Phys. J. B. 58 (2007) 423–431. doi:10.1140/epjb/e200700254-8.
ED
[25] M. Francoeur, M.P. Mengüç, R. Vaillon, Coexistence of multiple regimes for near-field
PT
thermal radiation between two layers supporting surface phonon polaritons in the infrared, Phys. Rev. B. 84 (2011) 75436. doi:10.1103/PhysRevB.84.075436.
CE
[26] O. Ilic, M. Jablan, J.D. Joannopoulos, I. Celanovic, H. Buljan, M. Soljacic, Near-field
AC
thermal radiation transfer controlled by plasmons in graphene, Phys. Rev. B. 85 (2012) 155422. doi:10.1103/PhysRevB.85.155422.
[27] X.L. Liu, Z.M. Zhang, Graphene-assisted near-field radiative heat transfer between corrugated polar materials, Appl. Phys. Lett. 104 (2014) 251911. doi:10.1063/1.4885396. [28] A.W. Rodriguez, O. Ilic, P. Bermel, I. Celanovic, J.D. Joannopoulos, M. Soljacic, S.G. Page 23 of 25
ACCEPTED MANUSCRIPT
Johnson, Frequency-selective near-field radiative heat transfer between photonic crystal slabs: a computational approach for arbitrary geometries and materials., Phys. Rev. Lett. 107 (2011) 114302. doi:10.1103/PhysRevLett.107.114302. [29] X.L. Liu, R.Z. Zhang, Z.M. Zhang, Near-field radiative heat transfer with doped-silicon
doi:10.1016/j.ijheatmasstransfer.2014.02.021.
CR IP T
nanostructured metamaterials, Int. J. Heat Mass Transf. 73 (2014) 389–398.
[30] J. Dai, S.A. Dyakov, S.I. Bozhevolnyi, M. Yan, Near-field radiative heat transfer between
AN US
metasurfaces: A full-wave study based on two-dimensional grooved metal plates, Phys. Rev. B. 94 (2016) 125431. doi:10.1103/PhysRevB.94.125431.
[31] X. Liu, Z. Zhang, Near-Field Thermal Radiation between Metasurfaces, ACS Photonics. 2
M
(2015) 1320–1326. doi:10.1021/acsphotonics.5b00298.
[32] B. Song, Y. Ganjeh, S. Sadat, D. Thompson, A. Fiorino, V. Fernández-Hurtado, J. Feist,
ED
F.J. Garcia-Vidal, J.C. Cuevas, P. Reddy, E. Meyhofer, Enhancement of near-field radiative heat transfer using polar dielectric thin films, Nat. Nanotechnol. 10 (2015) 253–
PT
8. doi:10.1038/nnano.2015.6.
CE
[33] S. Basu, B.J. Lee, Z.M. Zhang, Infrared Radiative Properties of Heavily Doped Silicon at
AC
Room Temperature, J. Heat Transfer. 132 (2010) 23301. doi:10.1115/1.4000171. [34] C. Luo, A. Narayanaswamy, G. Chen, J.D. Joannopoulos, Thermal radiation from photonic crystals: a direct calculation, Phys. Rev. Lett. 93 (2004) 213905. doi:10.1103/PhysRevLett.93.213905. [35] A.F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson,
Page 24 of 25
ACCEPTED MANUSCRIPT
Meep: A flexible free-software package for electromagnetic simulations by the FDTD method, Comput. Phys. Commun. 181 (2010) 687–702. doi:10.1016/j.cpc.2009.11.008. [36] A. Rodriguez, S.G. Johnson, Efficient generation of correlated random numbers using Chebyshev-optimal magnitude-only IIR filters, Eprint arXiv:physics/0703152. (2007).
CR IP T
http://arxiv.org/abs/physics/0703152 (accessed January 29, 2017).
[37] J.D. Joannopoulos, S. Johnson, J.N. Winn, R.D. Meade, Photonic crystals: molding the flow of light, 2nd ed., Princeton University Press, Princeton, New Jersey, 2008.
AN US
doi:10.1063/1.1586781.
[38] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994) 185–200. doi:10.1006/jcph.1994.1159.
M
[39] G. Parisi, P. Zilio, F. Romanato, Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method, Opt. Express. 20 (2012) 16690.
ED
doi:10.1364/OE.20.016690.
PT
[40] M. Francoeur, M. Pinar Mengüç, Role of fluctuational electrodynamics in near-field radiative heat transfer, J. Quant. Spectrosc. Radiat. Transf. 109 (2008) 280–293.
CE
doi:10.1016/j.jqsrt.2007.08.017.
AC
[41] D.L.C. Chan, M. Soljačić, J.D. Joannopoulos, Thermal emission and design in onedimensional periodic metallic photonic crystal slabs, Phys. Rev. E. 74 (2006) 16609. doi:10.1103/PhysRevE.74.016609.
Page 25 of 25
Meshed Doped Silicon Photonic Crystals for Manipulating Near-Field Thermal Radiation Mahmoud Elzouka , Sidy Ndao PII: DOI: Reference:
S0022-4073(17)30357-6 10.1016/j.jqsrt.2017.09.002 JQSRT 5828
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
10 May 2017 5 September 2017 6 September 2017
Please cite this article as: Mahmoud Elzouka , Sidy Ndao , Meshed Doped Silicon Photonic Crystals for Manipulating Near-Field Thermal Radiation, Journal of Quantitative Spectroscopy & Radiative Transfer (2017), doi: 10.1016/j.jqsrt.2017.09.002
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights
Significant enhancement in near-field thermal radiation is realized using practical and achievable micrometric gap, rather than nanometric gaps considered in majority of literature.
Near-field thermal radiation between meshed photonic crystal structure is investigated, for
CR IP T
the first time in literature. Enhancement in radiative heat transfer for meshed structures is 22 times higher than nonmeshed ones, which is promising for thermal rectification and thermal management applications.
First-principle method is used to simulate the random thermal vibrations of charges within
AN US
the material (i.e., the main mechanism of electromagnetic thermal emission).
Resonant modes validated and visualized by solving Helmholtz equation using weak-form
AC
CE
PT
ED
M
formulations.
Page 1 of 25
ACCEPTED MANUSCRIPT
Meshed Doped Silicon Photonic Crystals for Manipulating NearField Thermal Radiation Sidy Ndao, corresponding author University of Nebraska-Lincoln Mechanical and Materials Engineering Lincoln, Nebraska, 68588-0526, USA [email protected]
AC
CE
PT
ED
M
AN US
CR IP T
Mahmoud Elzouka University of Nebraska-Lincoln Mechanical and Materials Engineering Lincoln, Nebraska, 68588-0526, USA [email protected]
Page 2 of 25
ACCEPTED MANUSCRIPT
Meshed Doped Silicon Photonic Crystals for Manipulating NearField Thermal Radiation
CR IP T
Abstract The ability to control and manipulate heat flow is of great interest to thermal management and thermal logic and memory devices. Particularly, near-field thermal radiation presents a unique opportunity to enhance heat transfer while being able to tailor its characteristics (e.g., spectral
AN US
selectivity). However, achieving nanometric gaps, necessary for near-field, has been and remains a formidable challenge. Here, we demonstrate significant enhancement of the near-field heat transfer through meshed photonic crystals with separation gaps above 0.5 μm. Using a firstprinciple method, we investigate the meshed photonic structures numerically via finite-difference
M
time-domain technique (FDTD) along with the Langevin approach. Results for doped-silicon meshed structures show significant enhancement in heat transfer; 26 times over the non-meshed
ED
corrugated structures. This is especially important for thermal management and thermal
PT
rectification applications. The results also support the premise that thermal radiation at micro scale is a bulk (rather than a surface) phenomenon; the increase in heat transfer between two
CE
meshed-corrugated surfaces compared to the flat surface (8.2) wasn’t proportional to the increase in the surface area due to the corrugations (9). Results were further validated through good
AC
agreements between the resonant modes predicted from the dispersion relation (calculated using a finite-element method), and transmission factors (calculated from FDTD).
Page 3 of 25
ACCEPTED MANUSCRIPT
1. Keywords Near-field thermal radiation; photonic crystals; finite-difference time-domain; first-principle; cluster computing; meshed structures..
CR IP T
2. Introduction Manipulating heat transfer is of engineering interest for its potential applications in thermal storage, thermal management and thermal logic [1] and memory [2–4] devices. However,
AN US
controlling heat transfer is a challenge due to the lack of perfect thermal insulators, unlike in the case of electricity. Vacuum is an efficient thermal insulator, however not perfect since radiative heat transfer can still flow through vacuum. Nevertheless, using near-field effects, thermal radiation enhancement and/or suppression through vacuum can result in desired tailoring of heat
M
transfer.
ED
It has been demonstrated that near-field thermal radiation can enhance the radiative heat transfer dramatically, even beyond the classical theoretical limit of the blackbody radiation [5,6]. The
PT
transfer of thermal energy by near-field radiation occurs when a thermal emitter and receiver are brought very close to each other, at distances near or below the characteristic wavelength of
CE
thermal radiation. At this close proximity, in addition to propagating harmonic electromagnetic waves (as in the case of blackbody radiation), evanescent (i.e., non-propagating) waves which
AC
are confined to the thermal emitter’s surface (i.e., can only propagate along the surface) participate in the thermal energy transfer. The extra channels for heat flow created by these evanescent waves increase the rate of heat transfer exchange exponentially with decreasing separation gap (i.e., the smaller the gap, the stronger the evanescent fields are and the higher the thermal radiation exchange rate) [7,8]. These particularities of near-field thermal radiation make Page 4 of 25
ACCEPTED MANUSCRIPT
it ideal for thermal modulation applications, such as with thermal memory, thermal diode [9–13] and thermal switches [14,15]. Near-field thermal radiation has been of interest to scientists since the 1960s [16,17], and it has become increasingly an engineering research topic for the past two decades with advances in
CR IP T
nano/microfabrication techniques. Research efforts to enhance near-field thermal radiation range from the use of nano/microstructures [18–23], thin layers of polar dielectrics [24,25], patterned layers of graphene [26,27], extremely-thin photonic crystals [28] and metal-dielectric structures
AN US
that have hyperbolic dispersion [29–31]. Despite the numerous approaches, very few are
practical from the engineering standpoint; thin films of polar dielectrics [32] would be an example. Most significant engineering barriers include the fabrication of nanometric separation gaps and the implementation of exotic materials such as graphene.
M
In this paper, we are introducing a practical method for enhancing near-field thermal radiation,
ED
without necessarily using extremely small nanometer gaps. We propose using meshed photonic crystals with variable separation gaps as shown in Figure 1. The meshing increases the area
PT
available for heat transfer by thermal radiation and introduce new resonant electromagnetic modes. Both effects are expected to enhance the rate of radiative heat transfer. The meshed
CE
photonic crystals have a minimum separation gap of 0.5 μm, achievable with standard microfabrication techniques (i.e., projection lithography and deep reactive ion etching). We used
AC
gratings since they are one of the simplest structures and have commonly been used in optics applications. The photonic crystals are made out of doped-silicon which is fully compatible with most standard microfabrication techniques, well characterized, and most importantly, its optical properties can be controlled by varying its doping level [33]; it can be engineered to act as an opaque metal or transparent dielectric. We investigated the meshed photonic crystal numerically Page 5 of 25
ACCEPTED MANUSCRIPT
using a first principle finite-difference time-domain technique (FDTD). The technique solves Maxwell’s equations with added fluctuating current source (to simulate thermal radiation source). We investigate the effect of tooth depth and meshing on the heat transfer and discuss the resonant modes leading to the observed enhancement in heat transfer. In parallel, we calculated
CR IP T
the dispersion relation using finite-element method to verify the resonance frequencies in heat transfer from the FDTD results.
3. Methods
defined by its period , tooth height , and base thickness
AN US
The geometry of the meshed photonic crystal investigated in this study is shown in Figure 1. It is , spacing between meshed teeth
, separation distance
. We model the near-field thermal radiation using a first-principle
technique that simulate the source, scattering and absorption of thermal electromagnetic waves.
M
We use finite-difference time-domain (FDTD) method to solve Maxwell’s equations with
ED
random fluctuating current density source to represent the origin of thermal radiation. We adopted the same method introduced by Luo et. al. [34] to simulate the fluctuating current based
PT
on the Langevin approach. With this method, a simple variation is made to the FDTD algorithm [35] by introducing a randomly fluctuating component (in both magnitude and direction) to the
CE
polarization equation. The polarization equation defines the material’s polarization response due
AC
to local electric field, and it is used to update the value of the electric field in time stepping through the FDTD algorithm [34]
(1)
Page 6 of 25
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
is the vacuum permittivity. ,
CE
where
PT
Figure 1: Meshed photonic crystal layout (a) and computational domain (b-c) and
are the parameters of the Lorentz damped-
harmonic oscillator model that describes the polarization response to electric field.
AC
frictional coefficient or scattering rate, resonant frequency.
is the plasma frequency, and
is the
is the polarization
is a random variable added to the polarization equation to introduce
thermal fluctuations. In this work, we use doped-silicon which optical properties in the infrared regime (λ > 2 μm) can be modeled using the Drude model [33]. The photonic crystal structures
Page 7 of 25
ACCEPTED MANUSCRIPT
are modeled using heavily doped-silicon (p-type model parameters:
) with the following Drude
,
, and
.
To simulate the source of thermal radiation, the random fluctuations in the current density need
〈
〉
Where
(
)
is the current density in direction
(x, y, or z),
(2)
is the mean energy of Planck’s
AN US
oscillator;
CR IP T
to follow the fluctuation-dissipation theorem:
are Dirac delta functions, indicating that currents are uncorrelated in
M
and
(3)
both spatial space and frequency domain.
is the Kronecker delta which equal 1 for
ED
and zero otherwise, for an isotropic medium.
,
is the imaginary part of the relative permittivity
PT
of the material incorporating the fluctuating current, and
is the permittivity of free space.
To achieve fluctuations in current density that satisfies equation (2),
needs to have a
〉
(4)
AC
〈
CE
frequency profile of the form [34]:
where
is a constant comprising material’s parameters and volume of discretization elements
used in FDTD. The correlation in equation (4) is colored noise and it is possible to incorporate it in the FDTD algorithm. However, there are two inconveniences: the simulation will be only valid for a fixed temperature (used to calculate
Page 8 of 25
), and the generating colored noise needs
ACCEPTED MANUSCRIPT
more computation power and memory than the simple Gaussian white-noise [36]. Therefore here, we exploit the linearity of Maxwell’s equations and assume fluctuations in
as a white-
noise with flat spectral density (i.e., constant amplitude in frequency domain). The resulted solutions are scaled with respect to
to account for the temperature-dependent current
CR IP T
fluctuations.
Since the simulated structures have a discrete periodicity in x-direction and continuous
periodicity in z-direction, we simulate only one unit cell of the structure (as shown in Figure 1),
AN US
and infer the solution of the infinitely periodic structure using Bloch’s theorem [37]. Bloch’s theorem dictates Bloch periodic boundary condition at the boundaries normal to directions of periodicity (i.e., x and z directions), which describes the relationship between the fields (e.g.,
is the period or the length of the unit cell in x-direction, and
ED
where
M
electric field ) at the two adjacent sides as;
(5) is the Bloch
wavevector that dictates the phase shift between the two sides of the unit cell. The solution of the
PT
electromagnetic problem of the infinitely periodic structure in x direction is the superposition of
CE
all the solutions of a unit cell with Bloch periodic boundary condition, with all the values of that satisfy unique solutions (note that
of
gives the same solution). The range
which satisfies unique solutions is termed as Brillouin zone [37]. We can choose any range
AC
of
and
on condition that we have a span of
, so we have chosen
range to be
. Since our unit cell is symmetric about x axis, we expect that solutions with
identical to solutions with
. Therefore, we can solve our problem only for the range
Page 9 of 25
to be
ACCEPTED MANUSCRIPT
. Here, we assumed a constant value of values of
, and we didn’t solve for different
since they have a minimal effect on the spectral shape of the flux [18,28].
The boundaries of the unit cell normal to the y-direction are open boundaries with no reflection
CR IP T
to electromagnetic waves. To simulate an open boundary, we use an artificial perfectly matched layer (PML) [38] with thickness
and padding distance
, see Figure 1.
To calculate the net thermal radiative energy transfer from the emitter to the receiver, or the energy absorbed by the receiver, we calculate the difference between electromagnetic energy
AN US
crossing the two boundaries denoted by the dashed lines in Figure 1 b-d. The energy flux can be calculated by integrating the Poynting vector over the surface. The energy flux absorbed by the receiver (the bottom structure), due to randomly-fluctuating current distributed over the emitter (the upper structure) at a particular Bloch wavevector is denoted by
M
fluctuations in the simulation are generated from a white-noise
. Since the current
term, the actual heat flux
ED
corresponding to a certain Bloch wavevector should be represented by total heat flux is obtained by integrating over the
CE
∫
spanning over one Brillouin zone (i.e.,
PT
);
, and the
(6)
AC
In this study, we run the simulation with
step of
equal to the surface emissivity (or absorptivity) at far-field at ), and we named
. We scaled the
to be
(i.e., for a blackbody,
the spectral transmission factor. Since we incorporate
random values in our simulation, the results include noise which can be cancelled out by repeating the simulation over a number of times (~ 30) and averaging the results from all the Page 10 of 25
ACCEPTED MANUSCRIPT
runs. The number of averaged simulation runs depends on the running time for each individual run. In this work, we utilized a free open-source implementation of the FDTD algorithm called Meep (an acronym for MIT Electromagnetic Equation Propagation) [35]. We used finite-element method to investigate the dispersion relation of resonant modes created
CR IP T
between the toothed structures and to compare them to resonant frequencies and wavevectors from FDTD simulations. The method is based on converting the Helmholtz eigenvalue equation to a weak form, which can be solved using any finite-element technique. The details are
AN US
explained by Parisi et. al. [39]. This method incorporates the material dispersion and losses,
AC
CE
PT
ED
M
which resulted in accurate prediction of resonant frequencies.
Page 11 of 25
ACCEPTED MANUSCRIPT
4. Results and discussion 4.1. Effect of tooth height We investigate the effect of tooth height on the near-field thermal radiation. For transverse
and two photonic crystals designs with tooth heights of
and
for flat slabs
CR IP T
magnetic (TM) waves, Figure 2 shows TM waves transmission factors
and tooth spacing
. The transmission factor is calculated between two structures with spacing
.
For the sake of validation, we have plotted the transmission factor calculated analytically using
wave vector
can be estimated by ∑
AN US
Green’s function method [40] for the flat slab case; transmission factor corresponding to a Bloch , where
is an integer. Resonance
peaks appear on the corrugated structures for all values of the Bloch wavevector as seen on
M
Figure 2, unlike with the flat slab. The corrugated structures add new resonant modes which act as additional channels to tunnel photons from the emitter to the receiver. The -
-height
ED
toothed structure results in higher number of resonant modes in comparison to the -
-height
one at lower frequencies. This can be explained by taking a closer look at the magnetic field of
PT
resonant modes that is shown in Figure 2-b. The first three magnetic field profiles (A1, A2 and and -
-height
CE
A3) and the second three profiles (B1, B2 and B3) are corresponding to
toothed structures, respectively. The magnetic field profiles show waveguide-like modes of first,
AC
second and third orders, which can be distinguished by counting the number of nodes inside the cavity. Note that the fields don’t look uniform due to the losses incorporated in the prediction of the field profiles, an advantage of using the weak form formulation in solving the Helmholtz eigenvalue equation [39]. The waveguide-like modes are known to have a lower cut-off frequency (i.e., the frequency of the lowest order mode) for a larger cavity height. Therefore, the
Page 12 of 25
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
PT
Figure 2: Effect of tooth height on the transmission factor for TM waves
-height toothed structure shows resonant modes at lower frequencies than the -
AC
-
CE
a) spectral transmission factor for different tooth height at different Bloch wavevectors. Transmission factors for the flat slab are calculated analytically and represented by dotted lines. b) Plot of the magnetic field component in z direction for resonant modes at locations marked in (a) and marked in the dispersion relation in Figure 4-b. Mean energy of Planck’s oscillator is scaled to fit the plot. -deep
ones. It is also important to note from the plots the high transmission factors at infrared frequencies for the toothed structures at high values of Bloch wavevectors ( above) in comparison to the nearly zero transmission factor for the flat surface.
Page 13 of 25
and
ACCEPTED MANUSCRIPT
The additional resonant modes introduced by the corrugated structures and their prevalence at large Bloch wavevectors result in enhancement of the radiative heat transfer over the flat slab as shown in the integrated transmission factor
at the bottom of Figure 2-a and explicitly
supported by the total heat transfer plots in Figure 5-b. However, the enhancement in
CR IP T
transmission factors from these resonant modes applies only to TM waves. For transverse electric (TE) waves, there are no resonant modes since the doped-silicon has no magnetic
response, and the photon tunneling due to total internal reflection is dominant. The larger the surface area through which the frustrated total internal reflection occurs, the higher the expected
AN US
heat transfer. Therefore, for TE waves, the flat slab shows higher rate of heat transfer than the corrugated structures separated by
gap, as shown in Figure 5-a. This is because the flat
slabs have higher heat transfer surface area at small gaps than the corrugated surfaces. The case
ED
4.2. Effect of tooth meshing
M
is different when meshing occurs, as will be discussed in the next section.
It is intuitive from the experience with heat transfer at macro scale to expect the rate of heat
PT
transfer to be proportional to the surface area. This picture is valid as long as the heat transfer is a surface phenomenon (i.e., thermal radiation is emitted and absorbed by a small layer close to the
CE
surface). The case is different at microscale, where we need to consider the thermal radiation
AC
absorbed by and emitted from points across the entire material (i.e., bulk), such as in the case of the present meshed photonic structures. Figure 3-a shows the transmission factors for two corrugated structures with tooth height of 8 μm as a function of separation gap, dc, as shown in the schematic in Figure 3-b. Negative values of dc correspond to cases where meshing occurs. Figure 3-a shows a decrease in the number of resonant modes at low
values with decreasing
separation gap. This is due to the reduction in the volume available for the resonant waveguidePage 14 of 25
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 3: Effect of structures meshing on transmission factor.
CE
PT
(a) spectral transmission factor for different separation gaps at different Bloch wavevectors for TM waves. (b) schematic shows the toothed structures at four different separation gaps of -7.5, -2.5, 0.5 and 10 . (c) magnetic field profiles for the first three modes of fully meshed structures with . The eigenmodes locations are indicated in (a) and in Figure 4-c. like modes. However, the smaller the separation gap, the higher the transmission factor of the
AC
resonant modes at higher
values, and the stronger the resonant modes are; note the absence of
resonant modes for structures with a separation distance of
at
separation gap results in enhancement of the integrated transmission factor the bottom plot of Figure 3-a.
Page 15 of 25
. Decreasing the , as shown at
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 4: Dispersion relation for resonant modes between two toothed structures
ED
M
All structures have the same period and tooth separation . a) dispersion relation for , and . b) and c) are dispersion relations for , and and , respectively. The grayscale intensity of each point represents the ratio in field intensity across one unit cell. The filled contour plot in (c) represents the spectral transmission factor, and it shows the agreement between the resonance points and the peaks in the transmission factor. Here, the most important effect of meshing is the enhancement in transmission factors from non-
PT
resonant modes (i.e., total internal reflection) in comparison to the non-meshed cases; this
CE
enhancement is clearly observable at frequencies away from the resonance frequencies. This contribution from non-resonant modes increases with decreasing separation gap. As can be seen
AC
on the integrated transmission factors plot, the fully meshed photonic crystal ( has almost no resonance, while the resonance is more noticed for the case of
) .
Dispersion relation for the fully-meshed structures (Figure 4-c) shows a continuum of states without any gap for all frequencies of interest. The dispersion relation is similar to case of wave propagating in isotropic medium, or a layered medium with low contrast in refractive index. This Page 16 of 25
CR IP T
ACCEPTED MANUSCRIPT
AN US
Figure 5: Total heat transfer scaled to heat transfer of flat slabs separated by 7-μm gap, for a) TE, b) TM and c)TE+TM contributions. case is known to have no band gaps and can support waves with all frequencies. The case is different from the wave guide-like dispersion shown in Figure 4-b for two separated structure,
M
which has a couple of pseudo band gaps.
ED
4.3. Effect of separation distance on total heat transfer We calculate the total heat transfer
∫
between the toothed structures for different
PT
designs and multiple separation distances; results are shown in Figure 5 for TE, TM and TE+TM waves. In the figure, the heat transfer
aparts. Since there are no TE surface resonant modes for non-magnetic material
CE
surfaces
is scaled to the total heat transfer between two flat
AC
structures [41], such being the case here, the radiative heat transfer for TE modes is mediated by propagating modes or evanescent modes from frustrated total internal reflection. Therefore, the radiative heat transfer is dependent on surface area. This can be seen on Figure 5-a at the separation gap of
, where the flat surfaces outperform corrugated structures due to the
higher surface area available for frustrated total internal reflection to occur. Note also the similar heat transfer values for the two structures, tooth height of Page 17 of 25
and
, but same
at
ACCEPTED MANUSCRIPT
the separation gap of
. This similarity is the result of the two surfaces having the same
overlapped surface area. The heat transfer rates for structures with the ones with
are higher than
, further supporting that heat transfer from TE modes depend heavily
surfaces [30].
CR IP T
on the surface area available for total internal reflection and the separation between those
With TM modes (Figure 5-b) however; the structured surfaces outperform the flat surfaces at all separation gaps, by over an order of magnitude for the case of
and
of
and
but different
AN US
Comparing the heat transfer for the two structures having the same
.
, we find that the values of heat transfer are almost identical when the
structures are meshed, although the separation gap between the two surfaces is different. This shows that the effect of resonant modes on the heat transfer is much stronger than the effects of
M
frustrated total internal reflection.
and
ED
It is worth noting that the heat transfer from TE (TM) modes at the fully meshed case for is higher than the case of heat transfer between two flat surfaces by a factor of
. The enhancement in heat transfer is
PT
with separation gap
expected to be enhanced by a factor equal to the increase in the surface area resulted from
CE
corrugation (
), based on the classical view of thermal radiation (i.e., radiative
AC
heat flux is proportional to the area of surfaces exchanging the heat). However, the enhancement in heat transfer was
for TE (TM) modes, which is less than the augmented surface area
ratio of 9. This has two reasons; the heat flux per unit surface area between parallel infinite surfaces should be higher than the heat flux between surfaces with a finite length of the structure’s teeth). The second reason is that tooth width,
, is thinner than the
penetration depth of radiation. Overall, fully meshed structures achieved enhancement in Page 18 of 25
(i.e.,
ACCEPTED MANUSCRIPT
radiative heat transfer 26 times higher than the non-meshed structures separated This enhancement could result in thermal rectifications (
apart.
) as high as 2500% making
meshed photonic crystals ideal for thermal management applications such as thermal memory
5. Conclusions
CR IP T
and thermal diodes.
We have investigated meshed photonic crystals to enhance near-field radiative heat transfer between two structures at a minimum separation gap of
. The goal is to achieve
AN US
enhancement in radiative heat transfer at practical separation gaps other than the nanometric gaps suggested in the literature. We have solved for the radiative heat transfer using a first-principle method; FDTD with the Langevin approach to simulate the thermal fluctuating currents. The
M
enhancement in the heat transfer from non-meshed corrugated structures relative to a flat slab has been mainly attributed to TM resonant modes. In the case of meshed photonic crystals, further
ED
enhancement in heat transfer resulted from the contribution of non-resonant modes (i.e., frustrated total internal reflection), which increases proportionally with surface area. We
PT
calculated the dispersion relation for the meshed and non-meshed cases using finite-element
CE
method; calculated resonant frequencies were in agreement with the results from the FDTD simulations. The dispersion relations for the non-meshed structures were similar to wave-guide
AC
resonant modes. In the case of meshed structures, the dispersion relation was similar to that of a homogenous media or layered media with low contrast in dielectric constant, which shows a continuum of states at all frequencies of interest to thermal radiation. Using meshed structures with a minimum separation gap of
, we were able to demonstrate enhancement in
radiative heat transfer as high as 26 times when compared to non-meshed structures. This
Page 19 of 25
ACCEPTED MANUSCRIPT
enhancement could result in thermal rectifications as high R=2500% (R = (Qmeshed – Qnon-meshed)/ Qnon-meshed ×100%); by incorporating the corrugated structures into NanoThermoMechanical rectifier [13], for instance. The high contrast in heat transfer between meshed and non-meshed cases makes meshed photonic crystals ideal for thermal management applications such as
CR IP T
thermal memory and thermal diodes.
Acknowledgment
This work was supported by the National Science Foundation (NSF) through the Nebraska
AN US
Materials Research Science and Engineering Center (MRSEC) (grant No. DMR-1420645). This work was completed utilizing the Holland Computing Center of the University of Nebraska, which receives support from the Nebraska Research Initiative. We would like to also thank Prof.
M
Christos Argyropoulos for the insightful conversations on electromagnetics modeling.
[1]
ED
References
N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, B. Li, Colloquium: Phononics: Manipulating
PT
heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84 (2012) 1045–1066.
[2]
CE
doi:10.1103/RevModPhys.84.1045. R. Xie, C.T. Bui, B. Varghese, Q. Zhang, C.H. Sow, B. Li, J.T.L. Thong, An electrically
AC
tuned solid-state thermal memory based on metal-insulator transition of single-crystalline VO2 nanobeams, Adv. Funct. Mater. 21 (2011) 1602–1607. doi:10.1002/adfm.201002436.
[3]
L. Wang, B. Li, Thermal memory: A storage of phononic information, Phys. Rev. Lett. 101 (2008). doi:10.1103/PhysRevLett.101.267203.
Page 20 of 25
ACCEPTED MANUSCRIPT
[4]
M. Elzouka, S. Ndao, Near-field NanoThermoMechanical memory, Appl. Phys. Lett. 105 (2014) 243510. doi:10.1063/1.4904828.
[5]
A. Narayanaswamy, S. Shen, L. Hu, X. Chen, G. Chen, Breakdown of the Planck
doi:10.1007/s00339-009-5203-5. [6]
CR IP T
blackbody radiation law at nanoscale gaps, Appl. Phys. A. 96 (2009) 357–362.
L. Hu, A. Narayanaswamy, X. Chen, G. Chen, Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law, Appl. Phys. Lett.
[7]
AN US
92 (2008) 133106. doi:10.1063/1.2905286.
B. Song, A. Fiorino, E. Meyhofer, P. Reddy, Near-field radiative thermal transport: From theory to experiment, AIP Adv. 5 (2015) 53503. doi:10.1063/1.4919048. R. St-Gelais, L. Zhu, S. Fan, M. Lipson, Near-field radiative heat transfer between parallel
M
[8]
structures in the deep subwavelength regime, Nat. Nanotechnol. (2016).
Z. Chen, C. Wong, S. Lubner, S. Yee, J. Miller, W. Jang, C. Hardin, A. Fong, J.E. Garay,
PT
[9]
ED
doi:10.1038/nnano.2016.20.
C. Dames, A photon thermal diode., Nat. Commun. 5 (2014) 5446.
CE
doi:10.1038/ncomms6446.
AC
[10] P. Ben-Abdallah, S.-A. Biehs, Phase-change radiative thermal diode, Appl. Phys. Lett. 103 (2013) 191907. doi:10.1063/1.4829618.
[11] C.Y. Tso, C.Y.H. Chao, Solid-state thermal diode with shape memory alloys, Int. J. Heat Mass Transf. 93 (2016) 605–611. doi:10.1016/j.ijheatmasstransfer.2015.10.045. [12] M.J. Martínez-Pérez, A. Fornieri, F. Giazotto, Rectification of electronic heat current by a Page 21 of 25
ACCEPTED MANUSCRIPT
hybrid thermal diode., Nat. Nanotechnol. 10 (2015) 303–7. doi:10.1038/nnano.2015.11. [13] M. Elzouka, S. Ndao, High Temperature Near-Field NanoThermoMechanical Rectification, Sci. Rep. 7 (2017) 44901. doi:10.1038/srep44901.
CR IP T
[14] W. Chung Lo, L. Wang, B. Li, Thermal Transistor: Heat Flux Switching and Modulating, J. Phys. Soc. Japan. 77 (2008) 54402. doi:10.1143/JPSJ.77.054402.
[15] W. Gu, G.-H. Tang, W.-Q. Tao, Thermal switch and thermal rectification enabled by nearfield radiative heat transfer between three slabs, Int. J. Heat Mass Transf. 82 (2015) 429–
AN US
434. doi:10.1016/j.ijheatmasstransfer.2014.11.058.
[16] E.G. Cravalho, C.L. Tien, R.P. Caren, Effect of Small Spacings on Radiative Transfer Between Two Dielectrics, J. Heat Transfer. 89 (1967) 351. doi:10.1115/1.3614396.
M
[17] S. Basu, Near-field radiative heat transfer across nanometer vacuum gaps: fundamentals and applications, 1st ed., Elsevier, Oxford, UK, 2016.
ED
http://www.sciencedirect.com/science/book/9780323429948 (accessed March 14, 2017).
PT
[18] J. Dai, S.A. Dyakov, M. Yan, Enhanced near-field radiative heat transfer between corrugated metal plates: Role of spoof surface plasmon polaritons, Phys. Rev. B. 92
CE
(2015) 35419. doi:10.1103/PhysRevB.92.035419.
AC
[19] X. Liu, B. Zhao, Z.M. Zhang, Enhanced near-field thermal radiation and reduced Casimir stiction between doped-Si gratings, Phys. Rev. A. 91 (2015) 62510. doi:10.1103/PhysRevA.91.062510.
[20] R. Guérout, J. Lussange, F.S.S. Rosa, J.-P.J.-P. Hugonin, D.A.R. Dalvit, J.-J.J.-J. Greffet, A. Lambrecht, S. Reynaud, Enhanced radiative heat transfer between nanostructured gold Page 22 of 25
ACCEPTED MANUSCRIPT
plates, Phys. Rev. B. 395 (2012) 180301. doi:10.1088/1742-6596/395/1/012154. [21] J. Lussange, R. Guérout, F.S.S. Rosa, J.-J. Greffet, A. Lambrecht, S. Reynaud, Radiative heat transfer between two dielectric nanogratings in the scattering approach, Phys. Rev. B.
CR IP T
86 (2012) 85432. doi:10.1103/PhysRevB.86.085432. [22] S.-A. Biehs, F.S.S. Rosa, P. Ben-Abdallah, Modulation of near-field heat transfer between two gratings, Appl. Phys. Lett. 98 (2011) 243102. doi:10.1063/1.3596707.
[23] A. Didari, M. Pinar Mengüç, Near-field thermal emission between corrugated Surfaces
doi:10.1016/j.jqsrt.2015.02.016.
AN US
separated by nano-Gaps, J. Quant. Spectrosc. Radiat. Transf. (2015).
[24] S.-A. Biehs, Thermal heat radiation, near-field energy density and near-field radiative heat
M
transfer of coated materials, Eur. Phys. J. B. 58 (2007) 423–431. doi:10.1140/epjb/e200700254-8.
ED
[25] M. Francoeur, M.P. Mengüç, R. Vaillon, Coexistence of multiple regimes for near-field
PT
thermal radiation between two layers supporting surface phonon polaritons in the infrared, Phys. Rev. B. 84 (2011) 75436. doi:10.1103/PhysRevB.84.075436.
CE
[26] O. Ilic, M. Jablan, J.D. Joannopoulos, I. Celanovic, H. Buljan, M. Soljacic, Near-field
AC
thermal radiation transfer controlled by plasmons in graphene, Phys. Rev. B. 85 (2012) 155422. doi:10.1103/PhysRevB.85.155422.
[27] X.L. Liu, Z.M. Zhang, Graphene-assisted near-field radiative heat transfer between corrugated polar materials, Appl. Phys. Lett. 104 (2014) 251911. doi:10.1063/1.4885396. [28] A.W. Rodriguez, O. Ilic, P. Bermel, I. Celanovic, J.D. Joannopoulos, M. Soljacic, S.G. Page 23 of 25
ACCEPTED MANUSCRIPT
Johnson, Frequency-selective near-field radiative heat transfer between photonic crystal slabs: a computational approach for arbitrary geometries and materials., Phys. Rev. Lett. 107 (2011) 114302. doi:10.1103/PhysRevLett.107.114302. [29] X.L. Liu, R.Z. Zhang, Z.M. Zhang, Near-field radiative heat transfer with doped-silicon
doi:10.1016/j.ijheatmasstransfer.2014.02.021.
CR IP T
nanostructured metamaterials, Int. J. Heat Mass Transf. 73 (2014) 389–398.
[30] J. Dai, S.A. Dyakov, S.I. Bozhevolnyi, M. Yan, Near-field radiative heat transfer between
AN US
metasurfaces: A full-wave study based on two-dimensional grooved metal plates, Phys. Rev. B. 94 (2016) 125431. doi:10.1103/PhysRevB.94.125431.
[31] X. Liu, Z. Zhang, Near-Field Thermal Radiation between Metasurfaces, ACS Photonics. 2
M
(2015) 1320–1326. doi:10.1021/acsphotonics.5b00298.
[32] B. Song, Y. Ganjeh, S. Sadat, D. Thompson, A. Fiorino, V. Fernández-Hurtado, J. Feist,
ED
F.J. Garcia-Vidal, J.C. Cuevas, P. Reddy, E. Meyhofer, Enhancement of near-field radiative heat transfer using polar dielectric thin films, Nat. Nanotechnol. 10 (2015) 253–
PT
8. doi:10.1038/nnano.2015.6.
CE
[33] S. Basu, B.J. Lee, Z.M. Zhang, Infrared Radiative Properties of Heavily Doped Silicon at
AC
Room Temperature, J. Heat Transfer. 132 (2010) 23301. doi:10.1115/1.4000171. [34] C. Luo, A. Narayanaswamy, G. Chen, J.D. Joannopoulos, Thermal radiation from photonic crystals: a direct calculation, Phys. Rev. Lett. 93 (2004) 213905. doi:10.1103/PhysRevLett.93.213905. [35] A.F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson,
Page 24 of 25
ACCEPTED MANUSCRIPT
Meep: A flexible free-software package for electromagnetic simulations by the FDTD method, Comput. Phys. Commun. 181 (2010) 687–702. doi:10.1016/j.cpc.2009.11.008. [36] A. Rodriguez, S.G. Johnson, Efficient generation of correlated random numbers using Chebyshev-optimal magnitude-only IIR filters, Eprint arXiv:physics/0703152. (2007).
CR IP T
http://arxiv.org/abs/physics/0703152 (accessed January 29, 2017).
[37] J.D. Joannopoulos, S. Johnson, J.N. Winn, R.D. Meade, Photonic crystals: molding the flow of light, 2nd ed., Princeton University Press, Princeton, New Jersey, 2008.
AN US
doi:10.1063/1.1586781.
[38] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994) 185–200. doi:10.1006/jcph.1994.1159.
M
[39] G. Parisi, P. Zilio, F. Romanato, Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method, Opt. Express. 20 (2012) 16690.
ED
doi:10.1364/OE.20.016690.
PT
[40] M. Francoeur, M. Pinar Mengüç, Role of fluctuational electrodynamics in near-field radiative heat transfer, J. Quant. Spectrosc. Radiat. Transf. 109 (2008) 280–293.
CE
doi:10.1016/j.jqsrt.2007.08.017.
AC
[41] D.L.C. Chan, M. Soljačić, J.D. Joannopoulos, Thermal emission and design in onedimensional periodic metallic photonic crystal slabs, Phys. Rev. E. 74 (2006) 16609. doi:10.1103/PhysRevE.74.016609.
Page 25 of 25