Surface plasmon-phonon-magnon polariton in a topological insulator-antiferromagnetic bilayer structure

We present a robust technique for computationally studying surface polariton modes in hybrid materials. We use a semiclassical model that allows us to understand the physics behind the interactions between collective excitations of the hybrid system and develop a scattering and transfer matrix method that imposes the proper boundary conditions to solve Maxwell's equations and derive a general equation describing the surface polariton in a heterostructure consisting of N constituent materials. We apply this method to a test structure composed of a topological insulator (TI) and an antiferromagnetic material (AFM) to study the resulting surface Dirac plasmon-phonon-magnon polariton (DPPMP). We find that interactions between the excitations of the two constituents result in the formation of hybridized modes and the emergence of avoided-crossing points in the dispersion relations for the DPPMP. For the specific case of a $\mathrm{Bi}_2\mathrm{Se}_3$ TI material combined with a 3D AFM such as NiO, $\mathrm{MnF}_2$, or $\mathrm{FeF}_2$, the polariton branch with low frequency below 2 THz redshifts upon increasing the thickness of TI thin film, which leads to an upper bound on the thickness of the TI layer that will allow an observable signature of strong coupling and the emergence of hybridized states. We also find that the strength of the coupling between the TI and the AFM, which is parametrized by the amplitude of the avoided-crossing splitting between the two polariton branches at the magnon resonance frequency, depends on the magnitude of the magnetic dipole and the linewidth of the magnon in the AFM material as well as on the Fermi energy of Dirac plasmon in the TI. Finally, we show that materials with extremely high quality, i.e., low scattering loss rate, are essential to achieve an experimentally observable strong coupling between a TI and 3D AFM material. The overall analysis identifies the material properties that are necessary to achieve experimentally observable strong coupling for the interaction between THz excitations in a TI/AFM heterostructure and can thereby guide experimental efforts.

Figure 1

(a) Schematic of a multilayer structure consisting of N constituent layers that have the same width W along the $x$ direction. The $z$ axis is chosen as the growth direction of the structure. The thickness, permittivity, permeability in the $m\mathrm{th}$ layer, and optical conductivity of the carrier sheet at the $m\mathrm{th}$ surface/interface are denoted by $d_m$, ${\varepsilon }_m$, ${\mu }_m$, and ${\sigma }_m$, respectively, whereas $I_m$ indicates the interface matrix at the $m\mathrm{th}$ interface. (b) Schematic of the amplitudes of incoming and outgoing EM waves used in the scattering matrix approach. The EM wave is incident on the left surface in the figure.

Figure 2

The electric and magnetic components of the EM wave corresponding to the TE and TM polarization.

Figure 3

The dispersion of the bulk Dirac plasmon-phonon-polariton mode in $\mathrm{Bi}_2\mathrm{Se}_3$ (red curve) and bulk magnon polariton in $\mathrm{FeF}_2$ (blue line); the dashed black line represents the bare photon's dispersion relationship $\omega =ck$. The magnetization of the AFM material ($\mathrm{FeF}_2$) is along the $x$ axis.

Figure 4

(a) The structure under consideration in the remainder of this work. A TI is deposited on top of an AFM with arbitrary magnetization direction. An EM wave with, in general, both TE and TM polarization is incident on the top surface of the TI material to excite the electric degree of freedom in the TI thin film, which can then couple with the magnetic degree of freedom in the AFM layer. (b) A finite TI film with thickness $d_{\mathrm{TI}}$ in contact with a half-infinite AFM. $A_{i,j}$ and $B_{i,j}$, with $i\equiv x,y$ and $j\equiv 0,2$, are the amplitudes of the forward- and backward-propagating EM waves in the air (indicated by $j=0$) and within the AFM material (indicated by $j=2$).

Figure 5

Surface DPPMP dispersion in $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{FeF}_2$ structure calculated using Eq. (66) with the Fermi energy of the Dirac plasmon on its surface $E_F=1$ eV and the thickness of TI layer (a) $d_{\mathrm{TI}}=10$ nm and (b) $d_{\mathrm{TI}}=200$ nm.

Figure 6

Frequency-dependent dielectric function of topological materials for corresponding $\mathrm{Bi}_2\mathrm{Se}_3$ (blue), $\mathrm{Bi}_2\mathrm{Te}_3$ (red), and $\mathrm{Sb}_2\mathrm{Te}_3$ (green) with (a) the real part and (b) the imaginary part.

Figure 7

Surface DPPMP dispersion in the vicinity of magnon frequency calculated using Eq. (66) for (a) $\mathrm{Bi}_2\mathrm{Se}_3$/NiO, (b) $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{MnF}_2$, and (c) $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{FeF}_2$ bilayer structures. In all cases $d_{\mathrm{TI}}=10$ nm and $E_F=1$ eV.

Figure 8

Surface DPPMP dispersion in $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{FeF}_2$ bilayer structure calculated using Eq. (66) as a function of varying Fermi energy of the Dirac surface plasmon. $d_{\mathrm{TI}}=10$ nm and the in-plane wave vector $k_x=0.8\times {10}^4{\mathrm{cm}}^{-1}$ are kept constant.

Figure 9

Surface DPPMP dispersion in $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{MnF}_2$ structure calculated from Eq. (66) with $d_{\mathrm{TI}}=10$ nm, Fermi energy (a) $E_F=0.1$ eV, (b) $E_f=0.4$ eV, and (c) $E_f=1$ eV. The green dotted line in all figures is the dispersion of the bulk magnon polariton mode in the $\mathrm{MnF}_2$.

Figure 10

TI-AFM coupling strength in the $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{FeF}_2$, $\mathrm{Bi}_2\mathrm{Se}_3$/NiO, and $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{MnF}_2$ bilayer structures as a function of the Fermi energy of the Dirac plasmon in the TI film. In all cases $d_{\mathrm{TI}}=10$ nm. Note that we use the technique presented in our previous work [46] to extract the strength of the coupling between the TI and the AFM that we call $\mathrm{\Delta }$; the magnitude of $\mathrm{\Delta }$ is schematically depicted in Fig. 7. For each bilayer the magnitude of $\mathrm{\Delta }$ is extracted at the maximum anticrossing (around 1.6 THz for $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{FeF}_2$, at 1 THz for $\mathrm{Bi}_2\mathrm{Se}_3$/NiO, and around 0.26 THz for $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{MnF}_2$).

Figure 11

Surface DPPMP dispersion in $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{FeF}_2$ structure calculated using Eq. (66). For both calculations the Fermi energy of the Dirac plasmon is $E_F=0.7$ eV, the thickness of the TI layer is $d_{\mathrm{TI}}=80$ nm, and the linewidths of the $\alpha$ and $\beta$ phonons in the TI and the magnon in the AFM are set at the realistic values listed in Tables 1 and 2. The linewidth of the Dirac plasmon is (a) ${\mathrm{\Gamma }}_D=\frac{1}{{\tau }_{DP}}=10$ THz and (b) ${\mathrm{\Gamma }}_D=\frac{1}{{\tau }_{DP}}=0.1$ THz.

Figure 12

Surface Dirac plasmon-phonon-magnon polariton dispersion in $\mathrm{Bi}_2\mathrm{Se}_3$/$\mathrm{FeF}_2$ structure versus the variations of TI's linewidth (a) and AFM linewidth (b) calculated from Eq. (66) with the Fermi energy of Dirac plasmon on its surface $E_F=1$ eV and the thickness of TI layer $d_{\mathrm{TI}}=10$ nm at a fixed in-plane wave vector $k_x=0.13\times {10}^4{\mathrm{cm}}^{-1}$.