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Surface Phonon Polaritons (SPhPs) are hybrid light-matter waves that arise from the coupling of an electromagnetic wave with the optical phonons of the material. SPhPs are particularly interesting because they can confine light at the nanoscale, far below the diffraction limit. This confinement can lead to strong light-matter interactions, making SPhPs useful for a variety of applications.

Hexagonal boron nitride (hBN) is a van der Waals material that has gained significant attention in the field of nanophotonics due to its ability to support SPhPs in mid-infrared. hBN is a naturally hyperbolic material, meaning it exhibits hyperbolic dispersion for phonon polaritons. These hyperbolic SPhPs can propagate with in-plane wavevectors much larger than the wavevector of light in vacuum, leading to highly confined polaritons.

The in-plane permittivity of hBN is isotropic ($\varepsilon_{xx}=\varepsilon_{yy}$). Therefore, the SPhP propagates isotropically. In the landmark paper Peining Li et al., Infrared hyperbolic metasurface based on nanostructured van der Waals materials. Science 359, 892-896 (2018). DOI: 10.1126/science.aaq1704, the authors proposed a metamaterial consisting of nanostructured hBN. The nanostructure results in an effective medium with anisotropic in-plane permittivity and thus it supports unidirectional SPhP. In this notebook, we simulate and compare the SPhPs in a uniform hBN layer and a nanostructured hBN layer. Unidirectional SPhP is observed in the hBN metamaterial and the results are shown to be consistent with these in the publication.

As a side note, using anisotropic metamaterial as an effective medium has also been demonstrated to be an useful engineering technique in silicon integrated photonics. Please see the two case studies: waveguide crosstalk reduction and the broadband polarizer.

For more interesting contents in the realm of nanophotonics, please check out the case studies on Anderson localization, non-Hermitian metagratings, and plasmonic nanoantennas.

FDTD simulations can diverge due to various reasons. If you run into any simulation divergence issues, please follow the steps outlined in our troubleshooting guide to resolve it.

SPhP on Uniform hBN Slab¶

Simulation Setup¶

The frequencies of interest are 1405 1/cm, 1415 1/cm, and 1425 1/cm. In mid-infrared, 1/cm (wavenumber) is conventionally used while in Tidy3D the native frequency unit is Hz. We can convert 1/cm to Hz by multiplying td.C_0/1e4. The native length unit in Tidy3D is $\mu m$ so td.C_0 is in the unit for $\mu m$/s. Hence, there is an additional factor of $1e4$.

The in-plane and out-of-plane permittivities of the hBN are given by Lorentz oscillators with the parameter values taken from the referenced paper. The hBN is on a SiO$_2$/Si wafer. For SiO$_2$, the permittivity is also described by Lorentz while Si is considered to have a constant permittivity of 12.

The thickness of hBN is 20 nm and the thickness of SiO$_2$ is 250 nm.

We first investigate the SPhP in a uniform hBN slab. The system consists of three layers: a 20 nm hBN, a 250 nm SiO$_2$, and a thick Si substrate. Each layer can be defined as a Box.

To excite the SPhP, a PointDipole source 200 nm above the hBN top surface is used. The momentum of SPhP is much higher than that of the free-space photon at the same frequency. Therefore, SPhP can not be excited by sources like a plane wave due to the momentum mismatch. The PointDipole excitation contains a broad range of momentum, which can couple to SPhP.

To visualize the excited SPhP, we define a FieldMonitor 30 nm above the hBN top surface.

Next we define a Simulation using the previously defined components. A few points to note:

• The run_time needs to be sufficiently long to fully capture the propagation of SPhP.

• A fine grid needs to be used since SPhP has a small wavelength.

• The coupling efficiency of the PointDipole to the SPhP is relatively small so we also lower the shutoff threshold from the default 1e-5 to 1e-7 to ensure simulation accuracy.

• The system has two mirror symmetries in the x and y directions. Therefore, we set symmetry=(1,1,0), which reduces the simulation domain size by a factor of 4. Even though the simulation domain size is not large, due to the fine grid and long run_time, the simulation still requires a good amount of computation.

• We use Absorber instead of PML as the boundary condition since the simulation domain size is small compared to the wavelength. Using PML can also work in principle as long as the evanescent field does not leak into PML. For safety, we will just use Absorber.

Submit the simulation to run on the cloud.

Result Visualization¶

After the simulation is done, we will plot the real part of Ez to visualize the SPhP at various frequencies of interest.

For the uniform hBN slab, the SPhP propagates isotropically along the surface so we observe a circular wave front. As the frequency increases, the SPhP wavelength decreases.

SPhP on Nanostructured hBN Metamaterial¶

Simulation Setup¶

As a comparison, here we simulate the nanostructured hBN metamaterial, where hBN is patterned into periodic gratings with a subwavelength period. This metamaterial can be described by an effective permittivity tensor such that $$\varepsilon_\mathrm{eff, x} = \left( \frac{1 - f}{\epsilon_\mathrm{x}} + f \right) ^{-1},$$ $$\varepsilon_\mathrm{eff,y}=\varepsilon_\mathrm{y}(1-f)+f,$$ $$\varepsilon_\mathrm{eff,z}=\varepsilon_\mathrm{z}(1-f)+f,$$ where $f=\frac{w}{w+g}$ is the filling fraction. Following this, we can plot the permittivity of hBN as well as the effective permittivity of the nanostructured hBN in the case of 70 nm ribbon width and 100 nm period. For the hBN, $Re(\varepsilon_x)=Re(\varepsilon_y)<0$ between 1400 1/cm and 1425 1/cm, which is why it supports isotropic SPhP as observed in the previous simulation. For the nanostructured hBN, only $Re(\varepsilon_{eff,y})<0$ so in principle it supports unidirectional SPhP. Next, we will confirm it with simulation.

To define the grating structure, we simply use a for loop.

For this simulation, it's better to have a mesh commensurate with the period of the nanostructure in the $x$ direction. Therefore, we define a uniform grid with the grid size set to a fraction of the period. For the $y$ and $z$ directions, we simply use the same automatic nonuniform grid as before.

The simulation for the nanostructured hBN can be defined by copying the previous simulation and updating the structures and grid setting.

Result Visualization¶

After the simulation is complete, we plot the field distribution as before. Indeed we observe the unidirectional propagation of the SPhP in the $y$ direction supported by the hBN metamaterial, which is very distinctive from the SPhP in the uniform hBN slab. This result further validates the idea of nanoscale manipulation of light by geometric design. It opens up various possibilities for future nanophotonic engineering.