Near Field Radiative Heat Transfer around the Percolation Threshold

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Near Field Radiative Heat Transfer Around the Percolation Threshold in Al Oxide Layers Jaime E Pérez-Rodríguez, Giuseppe Pirruccio, and Raul P. Esquivel-Sirvent J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b01914 • Publication Date (Web): 09 Apr 2019 Downloaded from http://pubs.acs.org on April 9, 2019

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The Journal of Physical Chemistry

Near Field Radiative Heat Transfer around the Percolation Threshold in Al Oxide Layers

J. E. Pérez-Rodríguez, G. Pirruccio, and R. Esquivel-Sirvent* Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20364, México D.F. 01000, México E-mail: [email protected] ABSTRACT: We present a theoretical calculation of the near field radiative heat transfer (NFRHT) around the percolation threshold of oxide layers that forms on aluminum surfaces. As this layer grows the volume fraction of 𝐴𝑙 decreases and that of 𝐴𝑙2𝑂3 increases. During this process, the Al plasmon response degrades and the surface phonon modes of 𝐴𝑙2𝑂3 become active, thus increasing the spectral radiative heat flux. This occurs when the content of aluminum oxide on the surface approaches 67% in volume, which corresponds to the percolation threshold according to the Bruggeman effective medium theory. This oxidation fraction defines two regions: for volume fractions below the percolation threshold the total heat flux is lower than that of unoxidized Al, otherwise it is larger. Thus, oxidation considerably affects both the spectral and total radiative heat transfer.

INTRODUCTION In the near-field, Stefan-Boltzmann’s predictions related to the radiative heat transfer between two bodies at different temperatures are no longer valid due to the presence of evanescent optical modes1. One of the most striking consequences are the dependence of the radiative heat flux on the distance between the bodies2 and its high coherence3. As such, the NFRHT can be several orders of magnitude larger

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than in the far-field regime and even larger than the black-body prediction, as first measured by Hargreaves4. Conventionally, Wien´s thermal wavelength marks the boundary between the far-field and near-field in which the contribution of the evanescent waves become dominant5,6. There is plenty of experimental evidence from different groups that have successfully measured anomalous heat transfer at the nanoscale7-17 and showed enhanced NFRHT down to nanometers and even subnanometer separations18-22. The possibility of tailoring and controlling the NFRHT have attracted great interest due to its potential for various technological applications ranging from the micro- to the nanometer scale23,24. NFRHT tunability can be particularly useful in micro and nanoengineering, where thermal transistors25,26, thermal diodes27-29, nanophotonic thermal devices30, and heat-assisted data storage devices31 have been discussed. Controlling or modifying the NFRHT can be achieved by changing the dielectric function of the materials32,33. The dependence of the NFRHT on the dielectric properties and on the separation between the bodies has been experimentally observed in plate-plate4,34-36, sphere-plate37,38, and tip-plate39-40 configurations. Furthermore, the application of an external magnetic field induces an anisotropy in the permittivity tensor allowing the excitation of magneto-plasmons that changes the NFRHT41. This same principle has been used to control the NFRHT between two nanospheres42 and between arrays of nanoparticles giving rise to a photon thermal Hall effect43. Dynamic modulation of heat transfer is also possible by applying an external mechanical pressure in microdevices44. Other alternatives to modify the NFRHT include the use of phase change materials45,46 and composite systems, which constitute a way to tune the heat transfer at the nanoscale47,48. In these references, the dispersion relation of surface modes is modified by spoof plasmons or by the hybridization of plasmon and phonon polariton modes49-51. Among the many materials that are used in nanodevices, aluminum (Al) has gained a lot of interest52,53, in particular in plasmonics, since the wavelength region of light that can be accessed extends to the ultraviolet (UV). Many applications have been proposed, including inactivation of bacteria in the UV54, surface enhanced Raman spectroscopy using 𝐴𝑙 oxides55, deposition of Al films on nanosphere


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substrates56, modification of the absorption of graphene57, enhancement and modification of the absorption and fluorescence of dyes58 . A strong motivation for studying aluminum in the context of NFRHT is its low cost and ease of micro- and nanostructuring. The high electron density and large plasma frequency of Al, responsible for its unique plasmonic response59, makes it a poor thermal element when referring to NFRHT. However, Al surfaces suffer from a fast oxidation leading to a strong modification of the optical properties of Al. Experiments conducted on Al cylindrical nanoparticles show that for pure Al, i.e., 0% oxidation, a sharp plasmonic peak in the scattering efficiency at a wavelength of 370 nm is observed. When the oxidation reaches 50%, the plasmonic response is already degraded. Oxidation forms a few nanometer-thick Al2O3 layer. This has been demonstrated by X-ray photoelectron spectroscopy60,61. Since a well-defined layered system is formed, and the dielectric function of the oxide layer can be described by an effective medium approximation, it is possible to carry out the optical characterization of the Al2O3/Al system. In this paper, we present a theoretical calculation of the NFRHT between two Al surfaces held at different temperatures, and we show that the oxide layer that forms on the surfaces dramatically changes both the spectral and the total radiative heat flux. The analytical calculations are based on fluctuation electrodynamics, the effective surface impedance method and the transfer matrix method. The oxide layer is modeled by the Bruggeman effective medium theory. We observe a change in the NFRHT which occurs near the percolation threshold of the oxide layer and corresponds to the appearance of the surface phonon polaritons modes.

THEORY The theoretical description of the NFRHT is based on Rytov's theory of fluctuation electrodynamics6, that describes the heat flux radiated by thermally excited electromagnetic fields in bodies at a temperature T. The solution of Maxwell's


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equations, assuming fluctuating sources, relates the temperature of the body with the electromagnetic energy flux, in terms of the polarization of the electromagnetic waves (p or s). For two ideal (no roughness), parallel surfaces at temperature 𝑇1 and 𝑇2 separated by a distance 𝐿, the heat flux is given in terms of the average Poynting vector normal to the surfaces, i.e. 𝑄 =< 𝐸(1) × 𝐻(1) >∙ 𝑛 ― < 𝐸(2) × 𝐻(2) >∙ 𝑛, where 𝐸(𝑖) and 𝐻(𝑖) are the thermally excited electric and magnetic field in either slab, 𝑖 = 1, 2 and n is the vector normal to the surfaces. In terms of the temperatures 𝑇1 and 𝑇2 of each plate, the frequency 𝜔 and the parallel wave vector component 𝛽, the heat flux is1,2 𝑄 =

∫

∞

𝑑𝜔[Θ(𝜔,𝑇2) ― Θ(𝜔,𝑇1)]

0

∑∫

𝑗 = 𝑝,𝑠

∞

𝑑𝛽 0

(1

1

𝑝𝑟𝑜𝑝 (𝐺𝑒𝑣𝑎𝑛 ) 𝜔, 𝑝,𝑠 (𝜔,𝛽) + 𝐺𝑝,𝑠 (𝜔,𝛽)) = ∫𝑑𝜔𝑆

2

(2𝜋)

where the sum is over the two possible polarizations of light (p or s), Θ(𝜔,𝑇) is the Planck distribution function Θ(𝜔,𝑇) =

ℏ𝜔 ℏ𝜔 + ℏ𝜔/𝑘 𝑇 𝐵 2 𝑒 ―1

(2)

𝑝𝑟𝑜𝑝 and 𝑆𝜔 is called spectral heat function. The functions 𝐺𝑒𝑣𝑎𝑛 𝑝,𝑠 (𝜔,𝛽) and 𝐺𝑝,𝑠 (𝜔,𝛽),

appearing in 𝑆𝜔, are called energy transmission coefﬁcients and give the contribution to the NFRHT of the evanescent and propagating waves. These functions are

𝐺𝑒𝑣𝑎𝑛 𝑝,𝑠 (𝜔,𝛽)

4𝛽 𝐼𝑚(𝑟𝑝,𝑠)2𝑒 ―2|𝜅|𝐿 = , |1 ― (𝑟𝑝,𝑠)2𝑒 ―2𝑖𝜅𝐿|2

(3)

and 2

𝐺𝑝𝑟𝑜𝑝 𝑝,𝑠 (𝜔,𝛽)

=

𝛽(1 ― |𝑟𝑝,𝑠|2)

(4) ,

|1 ― (𝑟𝑝,𝑠)2𝑒 ―2𝑖𝜅𝐿|2


4

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where they have been written in terms of the wave vectors components parallel (𝛽)

(

and perpendicular 𝜅 =

𝜔 2

() 𝑐

― 𝛽2

)

to the plates. For the propagating waves

between the plates we have (𝜔/𝑐)2 ― 𝛽2 > 0 and for the evanescent case (𝜔/𝑐)2 ― 𝛽2 < 0. The dielectric and geometrical properties of the material are contained in the reflectivities 𝑟𝑝,𝑠. The analytical expressions for rp,s are given in the supplemental material of Ref. (33). In the limit of large separations and for a black body 𝑟𝑝,𝑠→0, the spectral function 𝑆𝜔 is the Planck distribution function and Eq.(1) gives the StefanBoltzmann law: 𝑄 = 𝜎(𝑇41 ― 𝑇42 ), being 𝜎 the Stefan-Boltzmann constant. We stress that the calculation of Q has to take into account both propagating and the evanescent modes of the nanocavity formed by the two oxidized surfaces because the distance L is comparable to the decay length of the evanescent modes33. In this work, we consider the system shown in Figure 1. It consists of two Aluminum-coated dielectric slabs separated by a gap L=50nm, kept at temperatures 𝑇1 = 400𝐾 and 𝑇2 = 300𝐾 respectively. Each slab is made of 𝑆𝑖𝑂2 described by a constant permittivity 𝜀𝑆𝑖𝑂2 = 3.9. The 𝐴𝑙 layer has a thickness of 𝑑𝐴𝑙 = 55𝑛𝑚 with a top oxide layer (AlOx) of thickness 𝑑𝐴𝑙𝑂𝑥 = 5𝑛𝑚.

Figure 1. Schematic of the system showing Aluminum-covered silica separated by a vacuum gap of distance L. The Al surface is allowed to oxidize forming an oxide layer


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of thickness 𝑑𝐴𝑙𝑂𝑥 = 5𝑛𝑚. In all the calculations, it is assumed that 𝑇1 = 400𝐾 and 𝑇2 = 300𝐾.

Dielectric function of 𝐴𝑙2𝑂3 ―𝐴𝑙 system The dielectric function of Al 𝜀𝐴𝑙(𝜔) has been measured in several studies and accurate values are available in literature62. The intraband transition region of Al is well described by a Drude model with a plasma frequency of 𝜔𝑝 = 14.75 𝑒𝑉 and a phenomenological damping constant 𝛾 = 0.082 𝑒𝑉. For 𝐴𝑙2𝑂3 the tabulated data63 is modeled by a sum of seven Lorentz oscillator 61 given

by 7

𝜀𝐴𝑙2𝑂3(𝜔) = 𝜀∞ +

∑𝜔

𝑗=1

2 𝑗

(5)

𝐴𝑗𝜔2𝑗 ― 𝜔2 ― 𝑖𝜔𝛾𝑗

,

with 𝜀∞ = 3.1, and 𝐴𝑗, 𝜔𝑗, and 𝛾𝑗 are the amplitude, frequency and damping of each resonance, respectively.

Aluminum surfaces are well known to oxidize in presence of oxygen. An oxidized surface can be described as a composite medium made of a volume fraction f of Al2O3 and 1-f of Al. Thus, the dielectric function of the oxide layer 𝜀𝐴𝑙𝑂𝑥, can be calculated using an effective medium approximation. For the system under consideration, Bruggeman’s theory is suitable because of the high values of f 64.

This approach has been also validated by the experimental measurements of

Knight et al.53 . 𝜀𝐴𝑙𝑂𝑥 is calculated by solving the Bruggeman equation

(1 ― 𝑓)

(

𝜀𝐴𝑙 ― 𝜀𝐴𝑙𝑂𝑥 𝜀𝐴𝑙 + 2𝜀𝐴𝑙𝑂𝑥

) ( +𝑓

𝜀𝐴𝑙2𝑂3 ― 𝜀𝐴𝑙𝑂𝑥 𝜀𝐴𝑙2𝑂3 + 2𝜀𝐴𝑙𝑂𝑥


)

= 0,

(6)

6

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Besides, this model correctly describes the degradation of the plasmonic response with the oxide fraction53, f. This is related to the transition from a conductor to an insulator of the oxide layer. We calculate the effective plasmon frequency of the oxide layer according to Ref. (65)

𝜔𝑝𝑒 = 𝜔𝑝 1 ―

3𝑓

(7)

2.

In Fig. (2) we plot 𝜔𝑝𝑒 as a function of 𝑓. The effective plasma frequency 𝜔𝑝𝑒 decreases as 𝑓 increases, showing the degradation of the plasmon modes and the metal-to-dielectric transition of the oxidized layer. At the percolation threshold of 𝑓~0.67, predicted by the Bruggeman theory, 𝜔𝑝𝑒 reaches zero. Equation 7 is only valid for f0.67, i.e., up to the percolation threshold.

Figure 2. Effective plasma frequency of the oxidized aluminum surface as a function of the oxide volume fraction f. The curve shows how the value of 𝜔𝑝𝑒 decreases until the value of f is large enough for the oxide layer to go from a conductor to an insulator.


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The variation of 𝜔𝑝𝑒 with f strongly influences the effective dielectric function given by Eq. 6. Figure 3 displays the imaginary part of the effective dielectric function of the oxidized aluminum top layer for different values of the oxidation. As the oxidation fraction increases the dielectric function goes from metallic to dielectric, passing through an intermediate phase in which the phonon modes of the oxide layer appear. The behavior switches around the percolation threshold.

Figure 3. Imaginary part of the effective dielectric function of the oxidized aluminum layer as a function of frequency. The curves are for the oxidation fractions (from top to bottom) f=0, .1,.2,.4,.5,.65,.75, 1. As the oxidation increases the metallic behavior is lost and the different phonons of the 𝐴𝑙2𝑂3 become active.

RESULTS


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First, the reflectivities in Eqs. (3) and (4) for the system air/AlOx/Al/SiO2 are calculated in terms of the surface impedance for both p- and s- polarized waves using the transfer matrix method66. This enables us to determine 𝐺𝑒𝑣𝑎𝑛 (𝜔,𝛽) for p𝑝 polarized waves (see Eq. (3)) as a function of the frequency and the parallel component of wave vector 𝛽. Figure 4(a) and (b) show the energy transmission coefficient for the Al surfaces without oxidation (f=0) and in case of a fully oxidized surface layer (f=1), respectively. Two important effects of the oxidation are observed. The first one is the shift of the gap surface plasmon-polariton (G-SPP) to lower frequencies due to the saturation of electrical charge in the aluminum by the oxygen. In the Supplemental Material the evolution of the G-SPP frequency as a function of f is further analyzed.


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Figure 4. Energy transmission coefﬁcient for p-polarization for the case of (a) unoxidized and (b) fully oxidized surface layer. The insets show the frequencies at which the surface modes of 𝐴𝑙2𝑂3

contribute to the heat transfer. The dashed

horizontal lines correspond to the three main phonon peaks in Fig. 3. The frequencies are normalized to 𝜔0 = 0.0658 𝑒𝑉 = 1014 𝑟𝑎𝑑/𝑠.

The second effect is the emergence of new hybrid surface plasmon-surface phonons polaritons due to the phonons of the 𝐴𝑙2𝑂3 layer present on the aluminum


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surface (see inset of Figure 4(b)). These modes are the result of the Fano interference between the broad gap surface plasmon-polariton, associated to the cavity formed by the aluminum surfaces, and the narrow surface phonons associated to the oxide layer32. It is worth noticing that the main resonance around 𝜔/𝜔0=1.9 is not related to a particular Lorentz oscillator in Eq. (5). It results from the superposition of the tails of all these resonances. The condition for Fano interference is caused by the proximity of each resonance and their damping parameter (𝛾𝑗). For convenience the frequency has been normalized to 𝜔0 = 0.0658 𝑒𝑉 = 1014 𝑟𝑎𝑑/𝑠. These new modes enhance the NFRHT, as shown in Fig. 5 (a) and (b), where we plot the spectral heat transfer for p- and s-polarization, respectively, for different values of the oxidation fraction. To highlight the role played by the oxidation, in Fig. 5 the heat flux is normalized to the one of unoxidized Al. For f=0, the main contribution to the NFRHT comes from the s-polarized waves as a consequence of the high plasma frequency of aluminum33. For an oxidation lower than the percolation threshold, there is no significant change in the spectral heat flux (see Supplemental Material). As the oxidation increases, we see that the resonant surface modes of the 𝐴𝑙2𝑂3 for p-polarized waves, lead to a large enhancement of the spectral heat function. On the other hand, for s-polarization, there is a decrease of the spectral heat flux compared to un-oxidized Al. The relevance of p-polarized waves for the oxidation surface is explained by Eq. (7): as f increases, the surface becomes more dielectric. Surface modes from dielectric surfaces are known to be mainly p-polarized.


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Figure 5. Normalized spectral heat flux for different cases of oxidation for (a) Ppolarization and (b) S-polarization. The flux is normalized to the flux of un-oxidized Al, 𝑟𝑎𝑑 𝑠

and 𝜔0 = 0.0658 𝑒𝑉 = 1014

and the gap separion is L=50nm.

The effect of oxidation is even more noticeable in Fig. 6, where the normalized total heat flux is calculated for different gaps and for the same oxidation fractions as in Fig. 5. In this Figure, we observed that below the percolation threshold and for gap separations 𝐿 < 100 𝑛𝑚, the normalized heat flux of the oxidized surface is smaller than the total flux of aluminum (dashed line). For small values of 𝑓, the surface phonon modes of 𝐴𝑙2𝑂3 are not active yet but there is enough oxide to partially screen the surface plasmons of the Al layer, thus decreasing Q. On the other hand, when f is larger than the percolation threshold the dielectric behavior of the oxide layer is sufficiently strong to allow surface phonon polaritons, resulting in the increased Q. When the separation is 𝐿 ≥ 100 𝑛𝑚, the oxidation layer has no effect on the total heat flux, suggesting that the decay length of the surface modes of the oxide layer is less than the separation 𝐿.


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103 0.6 0.7 0.8 1

102

Q(f)/Q(0)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


101

100

10-1 100

101

L (nm)

102

103

Figure 6. Normalized total heat flux for different cases oxidation. The dashed line corresponds to the case of aluminum 𝑓 = 0 oxidation. The gap separation is L=50nm.

CONCLUSIONS In this work we show that the oxidation of an Al surface has a significant effect on the near-field radiative heat transfer. Surface passivation of Aluminum results in an oxide layer with a well-defined thickness of only a few nm. As the fraction of Al2O3 increases, the percolation threshold is reached which corresponds to the degradation of the plasmons modes and the activation of the surface phonon modes. For small separation L

Near Field Radiative Heat Transfer around the Percolation Threshold

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