Citation: | Chen S, Wu XH, Fu CJ. Active tuning of anisotropic phonon polaritons in natural van der Waals crystals with negative permittivity substrates and its application in energy transport. Opto-Electron Sci 3, 240002 (2024). doi: 10.29026/oes.2024.240002 |
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Supplementary information for Active tuning of anisotropic phonon polaritons in natural van der Waals heterostructures by negative permittivity substrates and its application |
(a) Schematic of α-MoO_{3}/substrate heterostructure, where the substrate is SiC or Au. (b) Real parts of the components of the dielectric tensor of α-MoO_{3} along the principal directions. The Reststrahlen bands are shaded in different colors.
(a) The minimum of k_{x} and k_{y} calculated by Eq. (10). The blue shaded regions and black arrows denote the reorientation of the IFC of PhPs. (b) The local enlargement of the region is marked by the yellow panel in (a). The thickness of the α-MoO_{3} slab is 100 nm.
Minimum k of the excited l = 0 mode along the [001] crystalline direction, i.e., y axis. (a) The permittivity of the substrate is purely real. Dashed lines indicate the LO phonon frequency along the x- (white), y- (green) and z- (purple) directions and solid lines represent the TO phonon frequency along the y- (green) and z- (purple) directions. The black arrows show the excitation region of reorientated polaritons, which are surrounded by black and blue dashed curves and are labeled as 1 and 2. (b) The effect of the dissipation of substrate on the k of reorientated polaritons. The region where reorientated polaritons can be excited is bounded by the blue dashed lines. The thickness of the α-MoO_{3} slab is fixed to 100 nm.
Dispersion of PhPs in α-MoO_{3} slab with a thickness of 100 nm placed on the top of different substrates: (a) air; (b) SiC; (c) Au along the x (Left) and y (Right) directions. Dashed lines indicate the LO phonon frequency and solid lines represent the TO phonon frequency along the x- (red), y- (black) and z- (purple) directions. The false color plot displays the imaginary part of the Fresnel coefficient, r_{p} (ω, k), obtained from the transfer-matrix method (see details in the Supplementary information), which illustrates the polaritonic dispersion.
Dispersion and corresponding Im(r_{p}) of PhPs in a 100-nm-thick α-MoO_{3} slab for air (a) and (c), SiC (b), and Au (d) substrates at different frequencies. At an incident frequency of 936 cm^{−1}, the principal axis of the hyperbola-like curves lays along the [100] direction (a) while it is along [001] direction (b) when adopting SiC substrate. Dashed curves are calculated analytically from Eq. (2), and the blue and yellow curves present l = 0 and l = 1 modes, respectively. (e–h) The simulated near-field distribution, Re (E_{z}), is excited by an electric dipole on the α-MoO_{3} slab. The insets are the corresponding dispersions obtained by the Fourier transform of the electric distribution Re (E_{z}).
Left: the normalized cross-sectional near-field distribution of the HPhPs propagating along the surface. Right: corresponding field profile at x = 0. Points A-D come from Fig. 5. Differently, (a) and (c) adopt air substrate while (b) and (d) correspond to SiC and Au substrates. Calculate the real part of x-component of the electric field distribution, Re (E_{x}) is shown for Points A, C, and D, while the real part of the y-component of the electric field distribution, Re (E_{y}) is presented for Point B. The E_{x} and E_{y} are normalized by the corresponding maximum value, and the x-axis and z-axis are normalized by the thickness of the film. The black solid lines denote the air / α-MoO_{3} interfaces and α-MoO_{3} / substrate interfaces.
The calculated real and imaginary parts of q_{ez} at ω = 936 cm^{−1} in (a) and (b), and ω = 989 cm^{−1} in (c) and (d). An imaginary part-dominated q_{ez} represents the decaying along the direction perpendicular to the interface while the real part-dominated one corresponds to phase oscillation.
Effect of substrate on NFRHT: (a) Schematic of NFRHT between two heterostructure systems composed of α-MoO_{3} slab with a thickness of t and practical substrates: SiC and Au. The vacuum gap width is d. The top structures is the emitter with a high temperature T_{1} (300 K) and the structure below the vacuum gap is receiver with a low temperature T_{2} (299 K). (b) The RHF (left) and relative enhancement coefficient η (right) for three kinds of structures as functions of vacuum gap width, d at thickness of α-MoO_{3} slab t =20 nm. The structures studied include (I) the individual α-MoO_{3} slab, (II) heterostructure consisting of α-MoO_{3} and SiC, and (III) heterostructure consisting of α-MoO_{3} and Au. The gay dotted line denotes the black body limit. (c) The enlargement of panel (b) for small gap size. According to the relative enhancement coefficient of SiC substrate, the whole region is divided into three parts: Region I (η ≈ 1); Region II annihilation (η < 1); Region III enhancement (η > 1).
(a) Spectral heat flux for a vacuum gap size d =1000 nm at a thickness of α-MoO_{3} slab t =20 nm. The SR II represents the inside of RBs, and the SR I and SR III denote the outside of RBs. (b) The heat flux from three spectral regions for the three structures.
In-plane PTC at frequencies corresponding to the peaks 1-6 labeled in Fig. 9(a) for the α-MoO_{3} /substrate heterostructure and suspended configurations.
(a) The radiative heat fluxes (left) and relative enhancement coefficient η (right) for three kinds of structures as functions of the thickness of α-MoO_{3} at a vacuum gap d =20 nm. (b) Spectral heat fluxes for three kinds of structures, at t = 20 nm. The dark blue and gray shaded region indicate the reorientation and annihilation of the l = 0 mode in α-MoO_{3} / SiC heterostructure, respectively. The red shaded region represents the excitation of the l = 0 mode in α-MoO_{3} / Au heterostructure.
In-plane PTC for α-MoO_{3} / SiC heterostructure (a–d); the individual α-MoO_{3} slab (e–h) at frequencies of the peaks 1–4 labeled in Fig. 11(b). (i–l) denote the attenuation length of the fundamental mode (l = 0). The arrows indicate the movement of dispersion curves when the α-MoO_{3} slab is placed on the top of SiC substrate. The white dashed curves indicate the symmetric and anti-symmetric branches of the coupled modes.
In-plane PTC for α-MoO_{3} / Au heterostructure (a) and (b); the individual α-MoO_{3} slab (c) and (d) at a frequency of the points 5 and 6 labeled in Fig. 11(b). The blue and yellow dashed curves represent the dispersion relation of the l = 0 and l =1 modes, respectively. The white curves denote the coupled symmetry and anti-symmetry modes.
NFRHT along the [001] crystalline direction for d = 20 nm. (a) Left: radiative heat flux varies with the thickness of the α-MoO_{3} slab for the cases with and without substrates. Right: the relative enhancement coefficient. (b) Spectral heat flux and total heat flux(inset) for the 10 nm-thickness α-MoO_{3} slab. (c) Heat flux from every RB. In-plane PTC at incident frequency of 760 cm^{−1} (RB 1) (d–f), 940 cm^{−1} (RB 2) (g–i) and 980cm^{−1} (RB 3) (j–l) for three structures: individual α-MoO_{3} slab, α-MoO_{3}/ Au and α-MoO_{3}/ SiC heterostructures.