Nonreciprocal optical properties of thermal radiation with SiC grating magneto-optical materials

Han Wang; Hao Wu; Zhiyuan Shen

doi:10.1364/OE.25.019609

1. Introduction

As one of the three fundamental modes of heat transfer, thermal radiation is a method that emits electromagnetic waves due to the stochastically fluctuating currents induced by the thermal motion of charge carriers, and its basic theory and applied study plays a significant role in thermal science and thermal technology. In present, due to the advent of nanotechnology, the study on the spectral control of thermal radiation with micro/nano scale materials has been the hot spot in the research field, such as absorption enhancement [1–4], thermal management [5–8] and radiative cooling [9], and energy harvesting [10–12]. However, most of all the micro/nano scale structure above is reciprocal material, that is, behave as reciprocity. Corresponding to the reciprocal material, the nonreciprocal materials can break the time reversal and detailed balance due to the special nonreciprocal effect, and have the characteristics that are not available in reciprocal materials, such as the surface directional emission spectrum is not necessarily equal to its directional spectral absorption. Note that the magneto optical materials are the significant source generating the nonreciprocal effect. Under the action of magnetic field, the atoms, or ions within the intrinsic magnetic moment of material will produce magnetic induction phenomenon, resulting in the orderly arrangement of the magnetic moment, which affects the transmission of light in its internal features, called magneto-optic effect.

Nonreciprocal material has received gradually attention due to its intriguing asymmetric optical features, making the promising candidates to approach the Landsberg limit [13]. The photovoltaic field of the famous scholar Martin A. Green [13] puts forward the use of nonreciprocal (magneto-optical) material in the best offset, and near zero emissivity, causing the incident and emission optical path separation to achieve higher power output, which can achieve the near zero entropy growth. Moreover, professor of stanford university Fan Shanhui [14] believe that nonreciprocal photonic structures represent an important emerging direction for control of thermal radiation as the nonreciprocal photonic crystal structure violate the principle of detailed balance. However, previous reports carried out their research only limited in terahertz frequencies [15–19], that is, locating in the far infrared and millimeter band, but the research of the optical and thermal radiation properties was rarely reported for magneto optical microstructure.

Near thermal radiation band, Zhu and Fan [14] study the optical properties of n-InAs photonic crystal structure, and present the nonreciprocal effect in absorptivity, which present one kind of thermal emitter to exploit nonreciprocal photonics to enhance the efficiency of renewable energy systems. Pipa et al. [20] theoretically investigated the spectral and angular characteristics of 1D semiconductor based magneto photonic crystal thermal emission in an external magnetic field. It is found that the doped-InAs have remarkable magneto optical effect, and the transmittance is larger than the absorptance. Moreover, the thermal radiation properties has evidence change under the magnetic field, which is an important task for developing space-saving controllable magneto photonic crystal based infrared sources. Hence, the above works are only focused on InAs, lack of the study on nonreciprocal optical properties for other magneto optical materials, and the applicability of nonreciprocal theory has no further discussion.

Since the nonreciprocal properties of some kinds of magneto-optical material have been presented in the far-infrared, the aim in this work is to find one kind of magneto-optical materials with the nonreciprocal effect in the mid-infrared region, which is more applicable to the thermal radiation band. Note that the SiC is one important semiconductor with wide applications in photovoltaic and functional ceramics. Hence, the study of its nonreciprical effect as the magnetic field applied is important. In this paper, simulation results show that as the magnetic field and incidence angle are respectively set at 10T and 30° or 60°, SiC has nonreciprocal properties in the mid-infrared region and maintains a high absorption rate. Moreover, the influence of magnetic field intensity, structural parameters and incidence angle to the infrared absorption of magneto-optical materials are discussed in details. It is worth noting here that the common used Drude model is not suitable for the dielectric constants of SiC, which differs from those in the optical manual. In this work, the Lorentz-Drude model is proposed for calculating the magneto-optical constants, which agrees well with the data in optical manual. Therefore, our proposed SiC structures could be used as an incident and emission optical path separation optical instrument.

2. Model

For any material, the general expression of the dielectric tensor is

ε = (\begin{array}{l} ε_{x x} ε_{x y} ε_{x z} \\ ε_{y x} ε_{y y} ε_{y z} \\ ε_{z x} ε_{z y} ε_{z z} \end{array})

When the magnetization vector M is parallel to the Z axis, that is the polar magneto-optical Kerr effect and the Faraday Effect, the dielectric constant is invariable after the Z axis rotated 90 degrees, due to the symmetry of C4.

ε^{'} = C_{4}^{- 1} ε C_{4} = (\begin{array}{l} ε_{y y} - ε_{y x} - ε_{y z} \\ - ε_{x y} ε_{x x} ε_{x z} \\ - ε_{z y} ε_{z x} ε_{z z} \end{array})

According to $ε = ε^{'}$ ,for isotropic materials, the dielectric constant tensor of semiconductor can be simplified as

ε = (\begin{array}{l} ε_{x x} ε_{x y} 0 \\ - ε_{x y} ε_{x x} 0 \\ 0 0 ε_{z z} \end{array})

Here, it is need to find the suitable dielectric constant (ε) of SiC. There are three dispersion formulas can describe the dielectric constant. The first is the Drude model, in which the dielectric constant can be expressed as the following,

ε (ω) = ε_{\infty} - \frac{ω_{p}^{2}}{ω (ω + i γ)}

Note that the Drude model is used to describe the dielectric constant of magneto optical materials, such as InAs, InP, and so on. The second is the Lorentz model, which is not introduced in here. The third model is Lorentz-Drude, which are introduced to describe the dielectric constant of magneto-optical materials in this work, as the following:

ε (ω) = ε_{\infty} (1 + \frac{ω_{p}^{2}}{ω_{T}^{2} - ω^{2} - i γ ω})

In Fig. 1, the dielectric constants of SiC are plotted as Drude model, Lorentz-Drude model and data in Palik [21]. From Fig. 1, it can be seen that the results of Lorentz-Drude model for both Real and Imaginary part agrees well with datas of Palik, while that of Drude model keep straight line and could not trap the usual and unusual dispersion characteristics of SiC. The Real part decreases as the wavelength increases, which shows the usual dispersion characteristics, and increases as the wavelength increases, showing a unusual dispersion characteristics. Hence, in addition to the resonant frequency (12.66-12.71μm) near the unusual dispersion of the relationship, the other wavelength ranges are normal dispersion relationship. For the imagery part, it increases rapidly, and achieves the peak at wavelength of 12.57μm. Moreover, the dielectric constants of SiC in Palik are only discrete data points, which are not applicable to the calculation of nonreciprocal properties. Hence, the Lorentz-Drude model can be suitable to calculate the dielectric constant of SiC.So the dielectric constant tensor can be expressed as:

\bar{ε} = (\begin{array}{l} ε_{\infty} (1 - \frac{ω_{p}^{2} (ω + γ i)}{ω [{(ω + γ i)}^{2} - ω_{c}^{2}] - ω_{T}^{2} (ω + γ i)}) \frac{i ω_{P}^{2} ω_{c}}{ω [{(ω + γ i)}^{2} - ω_{c}^{2}] - ω_{T}^{2} (ω + γ i)} 0 \\ - \frac{i ω_{P}^{2} ω_{c}}{ω [{(ω + γ i)}^{2} - ω_{c}^{2}] - ω_{T}^{2} (ω + γ i)} ε_{\infty} (1 - \frac{ω_{p}^{2} (ω + γ i)}{ω [{(ω + γ i)}^{2} - ω_{c}^{2}] - ω_{T}^{2} (ω + γ i)}) 0 \\ 0 0 ε_{\infty} (1 + \frac{ω_{p}^{2}}{ω_{T}^{2} - ω^{2} - i γ ω}) \end{array})

where,

ε_{\infty}

is the high-frequency limit permittivity as

ε_{\infty} = 6.38

,

ω_{T}

is the angular frequency of the incident wave as

ω_{T} = 1.50 \times 10^{14} r a d / s

,

γ

is the collision frequency of free electrons (

γ = 1 / τ

,

τ

is Relaxation time) as

γ = 4 \times 10^{11} r a d / s

.

ω_{p} = \sqrt{N e^{2} / ε_{0} m^{*}}

is the plasma frequency, in rad/s. The doping concentration can be expressed as

N = a T^{3 / 2} \exp (- E_{g} / 2 k T)

, and a is a coefficient related material, where the Boltzmann constant k = 8.625 × 10-5 eV/k, Eg is the energy gap in eV,

T

is the temperature which the structure operates,

e

is the elementary charge,

ε_{0}

is the free-space permittivity, in

c^{2} / m^{2} N

. The cyclotron frequency follows

ω_{c} = e B / m^{*}

, where

B

refers to the applied magnetic field intensity. So the doping concentration can be calculated as

N = 3.7 \times 10^{23} m^{- 3}

.

Fig. 1 Dielectric constant of SiC for real and imagery part.

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3. Model reliability verification

The basic structure of the SiC arrays on a metal substrate is showed in Fig. 2, which has three layers. The top optically layer consists of SiC and air, with a periodic array of gratings having the thickness, d1, width, W and the period between gratings, Λ. In the mediate layer, it is the SiC plate with thickness d2. And the film is deposited on substrate of the semi-finite perfect electrical conductor (PEC) mirror, with neglected thickness due to opaque and total reflection.

Fig. 2 Schematic of a photonic crystal structure for nonreciprocal effect.

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When a magnetic field B is applied along the z axis, the relative dielectric constant of SiC dielectric constant can be expressed as shown in Eq. (1). Due to the PEC is a perfect reflector, the transmittance is zero, the absorptance can be expressed as:

A = 1 - R

In order to check the reliability of the above model, the absorptivity is calculated and compared to that obtained in [14], for TM wave under ± 61.28° incidence angle and 3T magnetic field, which is present in Fig. 2. The relative structure parameters are $d_{1} = 1.981 μ m$ , $w = 3.2 μ m$ , $Λ = 7.24 μ m$ , $d_{2} = 0.485 μ m$ . It can be seen that the two results agree well. Note that only slight deviation exists due to the different simulation methods. In this work, commercial FDTD (Lumerical Solutions, Inc.) software package is used for numerical study, while the Rigorous coupled-wave analysis (RCWA) is used. Nevertheless, FDTD and RCWA are all the full-wave simulation methods, that is, obtain the results through solving the Maxwell equations, while the problem of data spillovers in multilayer structures often exists in RCWA. Note that slight deviation may due to the simplicity of electric field tensor or solving the transmission coefficient matrix for avoiding inverting the diagonal matrix for eliminating data spillovers. Hence, it can be assumed that the above model was reliable. In Fig. 3, it is shown that the absorptivity peaks are different under the same magnetic field but opposite incidence direction (_θ _{= ± 61.28°}), as the nonreciprocal material breaks the time reversal symmetry. If no magnetic field is applied, the absorptivity peak would coincide with opposite incidence direction.

Fig. 3 Comparison of the absorptance in this work with that in [14].

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4. Results and discussions

In the section 2, the permittivity tensor of magneto optical materials under magnetic field is given in Eqs. (1)-(6). To simplify the complex of computation, a unitary matrix transformation U is used to diagonalize the permittivity as below, i.e. treat the anisotropic dielectric tensor as the plane symmetric. The approximate dielectric constant can be expressed as:

{\tilde{ε}}_{D} {=U}^{*} \cdot \tilde{ε} \cdot U = (\begin{array}{l} ε_{xx} 0 0 \\ 0 ε_{yy} 0 \\ 0 0 ε_{zz} \end{array}) = (\begin{array}{l} n_{x x}^{2} 0 0 \\ 0 n_{y y}^{2} 0 \\ 0 0 n_{z z}^{2} \end{array})

The equivalent refractive index n of SiC under magnetic field B = 5T is plotted in Fig. 4. In Fig. 4, the real part and imaginary part of the refractive index nxx, nyy and nzz are present, which is magnified around the wavelength of 10 μm and the fluctuation can be observed. Due to the action of magnetic field, it makes magneto-optical materials anisotropy and dispersion. Due to the magnetic field along the z axis direction, it only affects the dielectric constant in x-y plane, namely for the dielectric function tensor $ε_{xx}$ and $ε_{yy}$ . Therefore it has no effect for the refractive index $n_{z z}$ in z direction, so that its refractive index keeps constant with no magnetic field.

Fig. 4 Equivalent refractive index n of SiC under magnetic field B = 5T.

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Then, the effect of magnetic field for magneto-optical material is analyzed by changing the magnitude of the magnetic field to 10T, with other parameters unchanged. The real part and imaginary part of the refractive index n of SiC are present in Fig. 5. It can be seen that the oscillations behave more severe, and the peak increased over the entire frequency range, due to the strength of the magnetic field enhanced. Note that the overlap of real part and imaginary part has a certain blue shift. Hence, the nonreciprocal effect and the absorption peaks occur toward to the lower wavelength direction, as the magnetic field intensity increases.

Fig. 5 Equivalent refractive index n of SiC under magnetic field B = 10T.

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In order to study the effect of structure parameters to the nonreciprocal absorptivity phenomenon, the absorptance of SiC optical structure is plotted in Figs. 6(a)-6(d), at different incidence angle (_θ _{= 0°~70°}). Note that the structure parameters include grating thickness d1 changing from 1.7 μm to 2.6 μm, and baseboard thickness d2 changing from 1.7 μm to 2.6 μm. Through changing the grating thickness d1 and baseboard thickness d2, we observe that the absorptivity is sensitive to the change of d2, especially for incidence angle less than 30°. In Fig. 6, it is shown that the structure parameters can affect the absorptivity peaks most obviously at d1 = 2.3 μm and d2 = 0.6 μm,that is the microcavity resonant effect being strongest. Hence, the suitable structure parameter has been determined that $Λ = 4 μ m$ , $w = 2 μ m$ , $d_{1} = 2.3 μ m$ , $d_{2} = 0.6 μ m$ , $θ = 30^{o}$ and $θ = 60^{o}$ 。

Fig. 6 Absorptivity of different thickness for two layers with variable incidence angle.

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Base on the above structure parameters, the absorption performance of SiC has been simulated as plotted in Fig. 7. From the above two graphs we can clearly find that under the action of the magnetic field, when the incident angle is θ = ± 30 °and ± 60 °, the absorption properties of the periodic structure of the SiC material in the opposite direction are shifted, and the corresponding wavelength and peak are inconsistent, because the SiC material under the action of the magnetic field led to the structure has a nonreciprocal, destruction of the structure of the time inversion symmetry. In Fig. 7(a), it can be seen that the absorption peaks exist near 10.15μm and 10.25μm, respectively for 30° and 60°. In the above chart, under the same magnetic field, the same positive and negative incidence angle, it can be found that the non-reciprocity in the same material at different wavelengths of intensity is different, showing a tendency that increase with the increase of wavelength. Although the results only covers from 10.1 to 10.3 μm, SiC with the above structure also has excellent absorption performance during 9 to 15 μm, which is not shown here.

Fig. 7 Absorptivity of SiC gratings with variable incidence angle under different magnetic field intensity (a) B = 5T, (b) B = 10T.

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As the magnetic field increase to 10T, the absorption performance of SiC has been simulated and the structure of the steady-state field energy distribution in the opposite direction, plotted in Fig. 7(b). From the above two graphs we can clearly find that under the action of the magnetic field, when the incident angle is θ = ± 30°and ± 60°, the nonreciprocal absorption properties of the periodic structure of the SiC material in the opposite direction are more obviously, and the absorption peak blue shift near 10.12 μm and 10.2 μm. By contrasting the above diagram, the increase of the applied magnetic field can make the refractive index of the material more intense, resulting in the absorption peak relative to the weak magnetic field absorption peak offset and absorption performance in the direction of the difference is more intense, which leads to material nonreciprocal performance is more intense. Furthermore, the absorptivity curve has a significant increase in the trend than that of 5T. On the other hand, the nonreciprocal effect seems more obvious at 30° than that at 60°, and the absorptivity peaks changed to be two for 30° at the same wavelength range, which means that the absorptivity peaks interval shorten with the smaller incidence angle.

The Electrical energy density, shown in Figs. 8(a)-8(d), gives information about the field cross-sectional distribution inside SiC gratings, and its impacts in the different angles performance. There is relatively stronger electrical field intensity on the top and interior of SiC nanostructure as −60° incidence angle than as + 60° incidence angle at the same energies, which shows the obviously nonreciprocal phenomenon, both for magnetic field applied B = 5T and B = 10T. As the PEC substrate is utilized, the surface plasmon polaritons (SPP) excited at SiC gratings and PEC interface can also show nonreciprocity in the dispersion relation, and thus can induce different altering in the phase-matching condition, and then in the absorpsion spectrum. The nonreciprocity of the optical response of the SiC gratings actually originates from the nonreciprocal dispersion relation of the system along transverse direction, as shown in Fig. 8. This can be seen comparing the contrast between the surroundings (air and substrate) and SiC gratings. A spatial concentration of the electric energy inside the SiC gratings suggests that the good absorption characteristics at incident light direction. In contrast, weaker field intensity is observed on the top of the SiC nanostructure at the back of the incidence direction, due to the less excited by incident light with less energy. There is also relatively stronger electrical field intensity on the top and interior of SiC nanostructure as −60 ° incidence angle than as + 60 ° incidence angle with the magnetic field intensity applied as 10T, which shows the obviously nonreciprocal phenomenon in Figs. 8(c) and 8(d).

Fig. 8 Contour plots along the z-y plane of the Electrical intensity of SiC with variable incidence angle under magnetic field B = 5T and 10T.

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Note that, the other important structure parameter is the filling ratio f, and f = W/Λ. Due to the filling ratio is related to the micro-cavity resonant effect of the material structure, the peak value in the same band will change with the structural filling rate. Here, we study the SiC material in the band of about 10 μm, which can make the material nonreciprocal by adding magnetic field B = 5T with incidence angle 30° and 60°, and analyze the effect of magnetic field strength and filling ratio on its nonreciprocal absorptivity strength.

By comparing the absorption properties of the grating structures with different filling ratios under the same magnetic field strength, we can see that with the decrease of the filling ratio, the non-reciprocity of the structure is stronger for 60° incidence angle, shown in Fig. 9. Hence, the filling ratio is important to the nonreciprocal effect, which provides a reference on optimal filling ratios for subsequent research.

Fig. 9 Absorptivity of SiC with variable filling ratios under magnetic field B = 5T, (a) f = 0.4, (b) f = 0.7.

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5. Conclusions

In summary, we have not only provided the nonreciprocal optical properties of SiC with different structure parameters, such as thickness of different layers, filling ratios, but also performed a systematic investigation further exploring and understanding light coupling, propagation, and absorption nature of SiC grating under variable incidence angle and magnetic field strength. Impressively, the properly designed geometrical factors are proposed, and the good nonreciprocal absorption properties of SiC in thermal radiation wavelength band are presented. It is found that the nonreciprocal effect is much obvious in infrared wavelength, and the nonreciprocal effect could adjust the absorption characteristic, thus be able to tune the absorption for the specific frequency of incident light, and lead to the different absorption for the opposite direction of incident light through the destruction of time reversal symmetry. Hence, the nonreciprocal effect of magneto optical materials in absorptivity properties has important use for the applications in infrared absorption regulation.

Funding

Natural Science Foundation of Jiangsu Province (BK20160999) and National Natural Science Foundation of China (NSFC) (51606093).

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Nonreciprocal optical properties of thermal radiation with SiC grating magneto-optical materials

Abstract

1. Introduction

2. Model

3. Model reliability verification

4. Results and discussions

5. Conclusions

Funding

References and links

Cited By

Figures (9)

Equations (8)

Optics Express