Thermal near-field energy density and local density of states in topological one-dimensional Su-Schrieffer-Heeger chains and two-dimensional Su-Schrieffer-Heeger lattices of plasmonic nanoparticles

We derive a general expression for the electric and magnetic part of the near-field energy density of $N$ dipoles of temperatures $T_1,...,T_N$ immersed in a background field having a different temperature $T_b$. In contrast to former expressions this inclusion of the background field allows for determining the energy density of heated or cooled isotropic dipolar objects within an arbitrary environment which is thermalized at a different temperature. Furthermore, we show how the energy density is related to the local density of states. We use this general expression to study the near-field enhanced energy density at the edges and corners of one-dimensional (1D) Su-Schrieffer-Heeger chains and 2D Su-Schrieffer-Heeger lattices of plasmonic InSb nanoparticles when the phase transition from a topological trivial to a topological nontrivial state is made. We discuss the robustness of these modes when adding defects and the possibility to measure the topological edge and corner modes.

Figure 1

Cutout of an infinite SSH chain with $t=\frac{\beta }{2}d$ and lattice constant $d$. Adjacent particles belonging to the same sublattice $A$ or $B$ are separated by distance $d$.

Figure 2

Band structure of an infinite chain with (a) longitudinal and (b) transversal polarized InSb particles with radius $R=100\mathrm{nm}$ and lattice constant $d=1\mu \mathrm{m}$. Here, we consider interactions between particles with a maximal distance of $20d$ to each other, only.

Figure 3

Real parts of the complex eigenfrequencies of the chain modes determined by Eq. (26) for (a) longitudinal and (b) transversal polarization as a function of $\beta$. The finite chain consists of 20 InSb NPs with radius $R=100\mathrm{nm}$ and lattice constant $d=1\mu \mathrm{m}$. For $\beta <1$ (blue circles) the Zak phase is ${\gamma }_{\perp /\parallel }=0$ and for $\beta >1$ (red crosses) the Zak phase is ${\gamma }_{\perp /\parallel }=\pi$.

Figure 4

Cutout of an infinite 2D SSH lattice consisting of four sublattices. The unit cell includes one particle $A, B, C, D$ of each sublattice. Furthermore, the high symmetry points of the first Brillouin zone are shown.

Figure 5

Band structure of the infinite 2D SSH lattice for op polarization in a quasistatic regime for (a) $\beta =1$ and (b) $\beta =1.3$ considering interactions up to next nearest unit cell.

Figure 6

Band structure of the infinite 2D SSH lattice for ip polarization in a quasistatic regime for (a) $\beta =1$ and (b) $\beta =1.3$ considering interactions up to next nearest unit cell.

Figure 7

For a finite 2D SSH lattice of 36 particles real parts of the complex eigenfrequencies of the lattice modes determined by Eq. (26) are shown as a function of $\beta$ for (a) ip and (b) op polarization. For $\beta <1$ (blue circles) the Zak phase is ${\vartheta }_{x/y}=0$ for all band gaps and for $\beta >1$ (red crosses) the Zak phase is ${\vartheta }_{x/y}=\pi$ for the first and third band gap. The green box indicates one of the corner modes at $\beta =1.3$.

Figure 8

Normalized absolute values of the dipole moments $|p_{A,B,C,D}^{\mathrm{op}}|$ of two op edge modes at frequency $1.746\times {10}^{14}\mathrm{rad}/\mathrm{s}$ when exciting a particle in the edge at the (a) bottom or (b) left and (c) a corner mode at the frequency $1.749\times {10}^{14}\mathrm{rad}/\mathrm{s}$ when exciting the particle at the left bottom corner for $\beta =1.3$ as found in Fig. 7.

Figure 9

Full spectral energy density $u_{\omega }$ in (Js/${\mathrm{m}}^3$) (longitudinal and transversal polarization) above an SSH chain of NPs for (a) $\beta =1.3$ and (b) $\beta =0.7$. Here we choose 10 particles with radius $R=100\mathrm{nm}$. The spectral energy density is calculated 300 nm above the NPs at the edge mode frequency $\omega =1.7493\times {10}^{14}$ rad/s for $\beta =1.3$. Black ($A$) and red ($B$) crosses mark the positions of the NPs of both sublattices.

Figure 10

Spectral energy density terms (a) $u_1$, (b) $u_2$, and (c) $u_3$ in (Js/${\mathrm{m}}^3$) for $\beta =1.3$. The other parameters are the same as in Fig. 9.

Figure 11

Spectral energy density in (Js/${\mathrm{m}}^3$) for (a) longitudinal and (b) transversal polarization. The parameters are the same as in Fig. 9.

Figure 12

Full spectral energy density in (Js/${\mathrm{m}}^3$) of a 2D SSH lattice for (a) $\beta =1.3$ and (b) $\beta =0.7$ for a lattice of 36 NPs (positions are marked via red crosses) at the edge mode eigenfrequency $\omega =1.75592\times {10}^{14}$ rad/s. All other parameters are the same as in Fig. 9.

Figure 13

Full spectral energy density in (Js/${\mathrm{m}}^3$) of a 2D SSH lattice for (a) $\beta =1.3$ and (b) $\beta =0.7$ for a lattice of 36 NPs (positions are marked via red crosses) at the corner mode eigenfrequency $\omega =1.7494\times {10}^{14}$ rad/s. All other parameters are the same as in Fig. 9.

Figure 14

Full spectral energy density in (Js/${\mathrm{m}}^3$) for $\beta =1.3$ for a chain of 10 NPs at the (a) edge mode frequency with a vacancy in the NP chain of the $A$ (top) and $B$ (bottom) sublattice. All other parameters are the same as in Fig. 9.

Figure 15

Full spectral energy density in (Js/${\mathrm{m}}^3$) for $\beta =1.3$ for a lattice of 36 particles (positions are marked via red crosses) at the (a) edge mode frequency $\omega =1.7559\times {10}^{14}$ rad/s and (b) at the corner mode frequency $\omega =1.74942\times {10}^{14}$ rad/s with an absent NP in the middle of the chain. All other parameters are the same as in Fig. 9.

Figure 16

Full spectral energy density in (Js/${\mathrm{m}}^3$) for $\beta =1.3$ for a lattice of 36 particles (positions are marked via red crosses) at (a) the edge mode frequency $\omega =1.7559\times {10}^{14}$ rad/s and at the (b) corner mode frequency $\omega =1.7494\times {10}^{14}$ rad/s. All other parameters are the same as in Fig. 9.

Figure 17

Real part of the eigenvalues $E$ for (a) transversal and (b) longitudinal modes as a function of the lattice constant for $N=300$ NPs and $\beta =1.3$. Furthermore the ratio $u_2^{\nu }/u_1^{\nu }$ with $\nu =\perp ,\parallel$ for the energy density 300 nm above the second NP $u_2$ and the first NP $u_1$.