Near-complete violation of Kirchhoff s law in thermal radiation in ultrathin magnetic Weyl semimetal films

The ability to break Kirchhoff s law is of fundamental importance in thermal radiation. Various nonreciprocal emitters have been proposed to break the balance between absorption and emission. However, the thicknesses of the nonreciprocal materials are usually larger than 1/10 times of the wavelength. Besides, the previous proposed nonreciprocal emitters are complex, thus they can hardly be fabricated in experiment to verify the Kirchhoff s law for nonreciprocal materials. In this paper, we investigate the nonreciprocal thermal radiation of the magnetic Weyl semimetal (MWSM) film atop of the metal substrate. It is found that the strong nonreciprocal radiation at the wavelength of 9.15 {\mu}m can be achieved when the thickness of the MWSM film is 100 nm. The enhanced nonreciprocity is attributed to the Fabry-Perot resonances. The results indicate that the MWSM film is the promising candidate to engineer the ultrathin and simple nonreciprocal thermal emitters. What is perhaps most intriguing here is that the proposed structure can be more easily fabricated in experiment to verify the Kirchhoff s law for nonreciprocal materials.

What is most important, complex structures are usually needed [11][12][13][14][15][16][17]. Although MWSMs possess intrinsic nonreciprocity, they have not been comprehensively 4 investigated to engineer nonreciprocal thermal emitters, especially for ultrathin and simple nonreciprocal thermal emitters. Grating structures and prism-coupling structures have been used to enhance the difference between absorption and emission of MWSMs [20,23]. Due to the fabrication technologies, these structures are not easy to be fabricated in experiment at present. The fabrication of single MWSM film has been achieved by several groups [29][30][31]. However, the nonreciprocal thermal radiation of single MWSM film has not been fully explored.
In this work, the nonreciprocal radiation of a single MWSM film atop of the metal substrate is explored. The results show that strong nonreciprocal radiation at the wavelength of 9.15 μm can be achieved when the thickness of the MWSM film is 100 nm. The enhanced nonreciprocity is attributed to the Fabry-Perot (FP) resonances.
Our results show that the MWSM film is the promising candidate to engineer the ultrathin and simple nonreciprocal thermal emitters. Besides, the structure is promising to be used to verify the Kirchhoff's law for nonreciprocal materials, since it can be easily fabricated in experiment [31].

Model
In this work, the proposed structure is shown in Fig. 1, where a MWSM film with a thickness of d1 is on the top of the silver (Ag) substrate. The relative permittivity of Ag is described by the Drude model, i.e., When each pair of Weyl nodes are separated along the y-axis in the momentum space, the relative permittivity tensor of the MWSM can be described as [20,23] 0 00 0 The detail expressions of the relative permittivity components can be found in Refs.
[21] and [24]. When a  is zero, the permittivity tensor is symmetric, thus obeying Lorentz reciprocity [35]. When a  is nonzero, the permittivity tensor is asymmetric, thus breaking Lorentz reciprocity [35]. When the temperature is 300 K, two components d  and a  are shown in Fig. 2 The plane of incidence is x-z plane, thus there is no polarization conversion between two linearly polarized waves. When a TM-(transverse magnetic, with the magnetic field along the direction of y-axis) polarized plane wave is incident with an angle θ, the spectral directional absorption and emission of the structure can be calculated by [26] ( ) Here, R(θ, λ) and R(-θ, λ) are the reflection for the incident angle of θ and -θ at the wavelength λ, respectively. The difference between emission and absorption is defined as e  =− , which measuring the nonreciprocal radiation. The transfer matrix method for calculating the reflection of the multilayer structures with nonreciprocal materials is presented in Ref. [17].

Results and discussion
The difference between the absorption and emission varying with the angle of incidence and wavelength for different thicknesses of the MWSM film d1 are shown 7 in Fig. 3. When the thickness is equal to or larger than 10 μm, the ultra-strong nonreciprocal radiation is located around the wavelength of 8.3 μm and the angle of incidence is 89 o . When the thickness is 1 μm, besides the wavelength of 8.3 μm, large difference between absorption and emission can take place at the wavelength around 10.6 μm, as shown in Fig. 3(d). When the thickness is equal to or smaller than 0.  When the thickness of the MWSM film is 10 μm and the angle of incidence is 89 o , the absorption and emission are respectively shown in Fig. 4(a). One can see that the difference between the absorption and emission can reach 0.9 at the wavelength of  Fig. 4(c). One can see that the field is strongly located at the interface between the air and the MWSM film. The field is stronger at angle of incidence of 89 o than that at angle of incidence of -89 o .
The stronger is the field, the smaller is the reflection. Therefore, the absorption is larger than the emission. Therefore, the nonreciprocity is strongly related with the permittivity at different wavelengths. ( ) Besides, it is noted that the absorption reaches its maximum when the thickness is 0.1 μm. Therefore, the strong nonreciprocal radiation shown in Fig. 7(a) is attributed to the excitation of FP resonances in the MWSM film. The distribution of magnetic field at the wavelength of 9.15 μm along the z-axis is plotted in Fig. 7(b).
The intensity of the incident magnetic field is set to be unity. When the angle of incidence is 68 o , the magnetic field is enhanced at the interface between the MWSM film and the Ag substrate, thus the absorption is large in this case. However, the magnetic field is smaller at the interface for angle of incidence of -68 o , indicating that most of the incidence wave is reflected. According to Eq. (2), the emission is small for angle of incidence of 68 o . At the wavelength of 9.15 μm, the absorption, emission, and the difference between them as functions of the angle of incidence and the thickness of the MWSM is shown in Fig. 8. According to Figs. 8(a) and 8(b), one can see that the first order FP resonance is strong, while the other orders are weak. Besides, strong absorption and emission are realized at large angle of incidence. As shown in Fig. 8(c), it is clear the nonreciprocal radiation is strong at the first order FP resonance. In addition, the large difference between absorption and emission can be realized when the angle of incidence is larger than 40 o . It is hard to realize strong nonreciprocal radiation at small angles of incidence.

Conclusions
In summary, the nonreciprocal thermal radiation based on MWSM film is show that the MWSM film is the promising candidate to engineer the ultrathin and simple nonreciprocal thermal emitters.