Far-Infrared Reflection from Heterostructures Made of Ultrathin Ferromagnetic Layers

Recent progress has been made in creating terahertz magnons using ultrathin ferromagnetic layers. Due to their short lifetimes, these can be difficult to measure. Here we detail a calculation that shows that infrared magnons can be detected using standard reflection and attenuated total reflection measurements from a thin-film heterostructure made up of alternating ultrathin magnetic and nonmagnetic layers. We use an entire-cell effective-medium calculation to find the magnetic permeability of the heterostructure and then use electromagnetic boundary conditions to calculate reflectivity as a function of frequency. There are appreciable dips in the reflectivity at infrared magnon resonance frequencies, for realistic material parameters. Moreover, the strong coupling of magnon photons indicates the possible use of 50 GHz–1 THz magnons in integrated signal-processing devices.

Figure 1

Reflection geometry of the layered structure. The $x$-$y$ plane is the plane of incidence for infrared radiation. The magnetic layered film has thickness $d$ and is placed on top of a semi-infinite substrate that extends $y>0$. The effective magnetization of the structure, ${\mathbf{M}}_{\mathrm{eff}}$, is directed along $z$. An externally applied magnetic field ${\mathbf{H}}_{\mathbf{0}}$ is also applied along $z$. The angle of incidence in air $\theta$ is the angle the wave vector of the incident wave ${\mathbf{k}}_1$ makes with the normal to the surface. Here we consider the case of an incident transverse electric (TE) beam so that the oscillating $\mathbf{H}$ field is in the $x$-$y$ plane. The inset shows a schematic of the first few even standing waves in an ultrathin film, which have some net moment that can couple with the impinging light.

Figure 2

The geometry for attenuated total reflection is similar to normal reflectivity (see Fig. 1), but includes a dielectric crystal prism placed at a height $d_{\mathrm{gap}}$ above the magnetic thin-film composite with thickness $d$. The wave vector of the incident light inside the prism is denoted $\mathbf{k}$, as opposed to the wave vector in the air gap ${\mathbf{k}}_1$ and the wave vector in the magnetic metamaterial ${\mathbf{k}}_2$.

Figure 3

The real part of the effective Voigt permeability ${\mu }_1=({\mu }_{xx}{\mu }_{yy}+{\mu }_{xy}^2)/{\mu }_{yy}$ of heterostructures as a function of frequency. The results for alternating four atomic layers of iron and two atomic layers of a nonmagnetic material is shown by the solid (blue) line (so called 4/2 superlattice). The results for 5/2 and 6/2 superlattices are shown by the long-dashed (red) line and the fine-dashed (green) lines, respectively. There are peaks in the Voigt permeability where the lowest standing wave in the ultrathin magnetic films is resonant. Insets show the form of the $n=0$ standing waves in the ultrathin iron films with six and four atomic layers, respectively.

Figure 4

The real part of the effective Voigt permeability ${\mu }_1$ of 20/2 heterostructures as a function of frequency. The peaks in this figure correspond to the $n=2$ standing wave in the 20-atom-thick iron film being resonant. Plots are shown for a pinning field, which is 25% (dot-dashed line), 15% (solid green line), and 5% (dashed red line) of the exchange field. The $n=2$ standing-wave profile is drawn in the two insets above for weak pinning ($H_{\mathrm{pin}}=0.05H_{\mathrm{ex}}$, left) and for strong pinning ($H_{\mathrm{pin}}=0.25H_{\mathrm{ex}}$, right). The shading in the right inset indicates the atomic layers that contribute to a net dynamic moment of the mode.

Figure 5

The reflectivity from a 343-nm-thick composite film with $\sigma ={10}^6\mathrm{S}{\mathrm{m}}^{-1}$ on top of a dielectric substrate (${ϵ}_3=4.5$) in (a) standard reflection and (b) ATR. In the ATR geometry, there is a 5-$\mu \mathrm{m}$ air gap between the film and a prism with $ϵ=3$. The angle of incidence is ${45}^{\circ }$ in air for standard reflection and in the prism for ATR. The same three heterostructures are considered as in Fig. 3, namely 4/2 (solid blue lines), 5/2 (dashed red lines), and 6/2 (fine-dashed green lines).

Figure 6

ATR reflectivity from a 4/2 heterostructure of varying thickness: 34.3 nm (thick dashed line), 343 nm (solid line), and semi-infinite (fine dashed line). ATR parameters are the same as those quoted in Fig. 5.

Figure 7

(a) ATR reflectivity—as a function of frequency—from a 323-nm-thick 4/2 heterostructure of varying conductivity: ${10}^7\mathrm{S}{\mathrm{m}}^{-1}$ (thick solid black line), ${10}^6\mathrm{S}{\mathrm{m}}^{-1}$ (solid blue line), ${10}^5\mathrm{S}{\mathrm{m}}^{-1}$ (thick-dashed line), and ${10}^4\mathrm{S}{\mathrm{m}}^{-1}$ (fine-dashed line). The ATR parameters are the same as those quoted in Fig. 5. In (b), the $\sigma ={10}^4\mathrm{S}{\mathrm{m}}^{-1}$ result is shown for the angle of incidence inside the ATR prism equal to $+{45}^{\circ }$ (fine-dashed line) and $-{45}^{\circ }$ (solid line) to show nonreciprocal effects that appear for low values of conductivity.

Figure 8

ATR electromagnetic field components inside the material as a function of distance $y$. Fields are shown just off resonance [left panels (a) and (c)] and on-resonance [right panels (b) and (d)]. The top panels show the real part of $E_z$ and the bottom panels show the real parts of $H_x$. A 323-nm-thick 4/2 heterostructure (shaded region) is considered with $\sigma ={10}^6\mathrm{S/m}$.