Fabricating Quasiperiodic Tilings with Thermal-Scanning Probe Lithography

1 Introduction

Quasicrystals exhibit long-range aperiodic order which distinguishes them from periodic crystalline materials. Since their discovery, a major research effort has been undertaken to understand their structure and properties.^{1, 2} Characterising the physical properties of atomic-scale quasicrystals poses several challenges, primarily due to the aperiodic arrangement of atoms, which complicates structure determination and requires advanced measurement techniques.³ In particular, the repertoire of quasicrystals available is limited to certain structures, such as icosahedral, decagonal, and dodecagonal systems.

Quasicrystals are grown with Czochralski, Bridgman, Floating Zone, or Self-Flux methods.^{4, 5} Growth conditions such as temperature, pressure, and composition, are carefully controlled to produce high-quality quasicrystals with desired properties. Characterising the properties of the bulk and surfaces of quasicrystals is performed with various diffraction, spectroscopic and microscopy techniques.⁶ Pseudomorphic systems of atomic overlayers on quasicrystal surfaces have also been grown.⁷ Overall, the preparation and characterisation of quasicrystals is challenging, spurring novel approaches.

Nanolithography techniques encompass artificial fabrication or manipulation of nanoscale structures to create patterns on a substrate. At the nano- to mesoscopic scale, nanolithography fabrication techniques open new possibilities for precisely engineering the size, shape, and arrangement of nanostructures, thus enabling tailored functionalities.^8-11 Quasicrystalline nanostructures have demonstrated exceptional properties with enhanced solar efficiency,^{12, 13} improved catalytic performance,¹⁴ and thermoelectric applications.^15-17

Quasiperiodic tilings model the surface of quasicrystals and have the potential to be fabricated with established nanolithography techniques.¹⁸ Broadly, there are three main nanolithography techniques described in the literature which are used to produce quasiperiodic tilings (see Table 1): scanning probe lithography (SPL), electron-beam lithography (EBL), and photolithography. In SPL a direct-write probe manipulates a substrate surface by rearranging, etching, or removing the atoms or molecules, either chemically or physically. Techniques such as thermal-scanning probe lithography (t-SPL) remove surface atoms or molecules by thermal sublimation with a heated probe.¹⁹ The resolution of SPL techniques is limited by the probe dimensions. EBL involves irradiating a surface with a focused beam of electrons through a mask covering specific areas of the substrate, or directly writing a pattern similar to SPL with e-beam resists. Resists are materials, typically polymeric, which are designed to break down under certain stimuli to transfer a pattern from a mask or directly onto a substrate. EBL resolution is limited by the de-Broglie wavelength of electrons interacting with the resist, although unlike a direct-write process such as SPL, charging and proximity effects need to be corrected for. Finally in photolithography a light-sensitive photoresist is patterned with incident photons, through a mask or a directly focused beam. With this technique, quasiperiodic Moiré patterns emerge with multiple interference beams with incommensurately modulated periodicity, spatial frequency, or angles.²⁰ The resolution is poorer than in SPL or EBL, due to the long wavelength of the photons employed in these techniques. Table 1 summarises the recent nanolithographic fabrication of quasiperiodic tilings, listing the techniques used and a brief description of the tilings explored and key results.

Table 1. Quasiperiodic tilings fabricated with various nanolithography techniques.

Technique	Description	Authors
t-SPL	12-fold reciprocal space etched grating for photolithography.	Lassaline, et al.21
EBL	10-fold Penrose22 and Fibonacci23 superconducting pinning arrays.	Kemmler, et al.22 and Villegas, et al.23
	12-fold air holes as an anti-reflection layer of a solar cell.	Mericer, et al.12
	10-fold Penrose tiling as an artificial spin-ice with disconnected edges.	Shi, et al.24
	Magnetic Properties of 10-fold Penrose and 8-fold Ammann-Beenker tilings.	Bhat, Farmer, Sung, et al.25-30
Photolithography	High-symmetry photonic quasicrystals fabricated with photolithography techniques from 7- up to 36-fold centred structures, with a range of applications and approaches.	Varadarajan and Shekar,13 Mahmood, et al.,20 Yang and Wangl,31 Vitiello, et al.,32 Xi and Sun,33 Langner, et al.,34 and Gao and Liu35

This paper aims to pedagogically demonstrate the associated challenges with fabricating quasiperiodic systems, and how to overcome these challenges, using a t-SPL technique. Results of successfully fabricated tilings, with a range of possible uses, are presented.