Terahertz metamaterials and systems based on rolled-up 3D elements: designs, technological approaches, and properties

Electromagnetic metamaterials opened the way to extraordinary manipulation of radiation. Terahertz (THz) and optical metamaterials are usually fabricated by traditional planar-patterning approaches, while the majority of practical applications require metamaterials with 3D resonators. Making arrays of precise 3D micro- and nanoresonators is still a challenging problem. Here we present a versatile set of approaches to fabrication of metamaterials with 3D resonators rolled-up from strained films, demonstrate novel THz metamaterials/systems, and show giant polarization rotation by several chiral metamaterials/systems. The polarization spectra of chiral metamaterials on semiconductor substrates exhibit ultrasharp quasiperiodic peaks. Application of 3D printing allowed assembling more complex systems, including the bianisotropic system with optimal microhelices, which showed an extreme polarization azimuth rotation of 85° with drop by 150° at a frequency shift of 0.4%. We refer the quasiperiodic peaks in the polarization spectra of metamaterial systems to the interplay of different resonances, including peculiar chiral waveguide resonance. Formed metamaterials cannot be made by any other presently available technology. All steps of presented fabrication approaches are parallel, IC-compatible and allow mass fabrication with scaling of rolled-up resonators up to visible frequencies. We anticipate that the rolled-up meta-atoms will be ideal building blocks for future generations of commercial metamaterials, devices and systems on their basis.

but with closed rings instead of split rings. Overlapping of the ends can be controlled both by the diameter (i.e. by thicknesses and strains of film layers) or by the lengths of the initial lithographic strips. It should be noted that the post-rolling metallization provides metal contact at the overlapping ends of the ring, while the ring rolled-up from metal-semiconductor film has a semiconductor layer in between the metal ring ends.
To make square lattices of right-handed (c) and left-handed (d) helices we used the mirror -image patterns (the pattern for right-handed helices is shown in Fig. 2a). These helices are rolled-up from the strained nanofilm similar to the one in Fig. 2b, but the film in Fig. 2b was rolled-up altogether with metal layers, while the helices in Fig. S2 c,d were covered with Pd after the rolling-up. It results in two-turn helices (Fig. S2 c,d) instead of one-turn helices (Fig. 2b) for the same length of the strip. The diameter of structure that was rolled up altogether with metal is twice more because of the metal layer elastically resists to the rolling-up of strained film.
In contrary, being covered with metal after the rolling-up the helix practically does not change its diameter, as far as the metal layer does not have internal strains in this case. Post-rolling metallization approach is especially important for fabrication of metamaterials and metasurfaces for IR and optical ranges, where rolled-up resonators must have submicrometer and nanometer diameters and therefore are to be rolled up from much thinner strained films. A number of methods (chemical solution deposition, atomic layer deposition (ALD), MO CVD and others) allow one to deposit metal layers on the both sides of rolled-up film elements without their deformations. The post-rolling metallization approach inherits such advantages of the rolling-up of pure semiconductor epitaxial films as the extreme accuracy both of diameter and the rolling-up direction. It should be noted that the post-rolling metallization also allows making a metal contact between the overlapping ends of 3D elements (Fig. S2 a,b).
The 3D post-rolling metallization usually covers the substrate with metal as well.
For applications where the metallized substrate is of hindrance it can be removed by transfer of rolled-up elements to another substrate or by embedding them into a free transparent polymer film as described in the main text.   Hasenohrl, Wet-etch bulk micromachining of (100) InP substrates. J. Micromech. Microeng. 14, 1205-1214 (2004)] in contrary to Si. This fact manifests itself in the lack of 4-fold symmetry of etching pits and therefore, of the rolled-up structures (Fig. S3, g). Here we stop the etching at the moment, when one set of parallel z-elements has been rolled-up completely and each element is freely suspended between the ends, while the perpendicular elements have not been rolled up completely and each element is additionally supported in the middle with a stem (Fig. S3 g).     Figure S7 a presents a schematic of the experiment further corroborating suggested mechanism of arising quasiperiodic peaks in the system of 4-fold symmetric lattice of helices on the substrate. When such system is placed between two diaphragms (diaphragm diameter is 1 mm) the second diaphragm restricts the number of round-trips in the substrate for the oblique waves that can be transmitted through (see schematic in Fig. S7 a). In accordance with suggested mechanism the peak-to-peak amplitude in polarization spectra decreases, and for the lower frequencies it decreases more than for higher frequencies (see where the square brackets denote the vector product. Here, we take into account the fact that the system of the polarizable elements is located in the plane 0 z h   . The wave vectors k  are also defined by formulas (S1).
From formulas (S1) and (S2) and from the first inequality in (1) it follows that for 0 n m   the system of polarizable elements emits a wave propagating along the z-direction.
In the case of | | | | 0 n m   , the system emits evanescent waves.
Under the adopted assumptions, we formulate the condition of wave reflection from the lower internal boundary of the dielectric slab as  ê The exponential factor is caused by the position of the lower boundary with the coordinate z h  .
To allow for the weak decay of the waves in the dielectric, we introduce the damping constants j  : where a l is the absorption length of the waves in the dielectric layer and  is the angle of inclination of the waves to the z -axis.
At the upper internal boundary of the plate, we have: Here, the first term is caused by the reflection of the waves at the upper boundary of the plate.
The second term in expression (S8) determines the contribution due to the waves emitted by the system of the polarizable elements. The third term is the contribution due to the wave falling Here, the expression with the summation symbol stands for the contribution due to the fields of the waves in the dielectric slab which leave the plate through its upper boundary; s E  is the strength of the electric field due to the polarizable elements that surround the selected polarizable element; and r E  is the strength of the electric field that arises as a result of the reflection of the polarizable-element-emitted waves from the upper face of the slab. The last term in (S11) is the contribution due to the wave falling onto the system together with the magnitude of the field due to the same wave reflected from the upper face of the slab; in this term, 0 where Г is the doubled damping ratio, ω 0 is the half-wave resonance frequency of the helix, ω 0 ≈2π*c/λ 0, and λ 0 =2*l 0 , where l 0 is the length of the unwound helix.
The results of the semi-analytical modelling of a square lattice of helices on GaAs substrate are presented in Fig. S9. It is important to note that the periods in the ORD and CD spectra are identical, and they differ from the period in the transmission spectra.