contiene menos del 0.5 por ciento (en masa) de carbono orgánico; y

To understand the unusual field distribution the HMM and utilize it properly in lithography, we considered a waveguide structure with HMM as its core and PR as claddings. Using 𝜖₁ for PR and 𝜖̿ for HMM as an effective medium, by matching the boundary conditions for 𝐻𝑦 and 𝐸𝑥 at the

interface 𝑧 = 0 with 𝜖𝑥𝑥∙ 𝜖1< 0, the corresponding 𝑘𝑥 with surface wave resonance point 𝑘𝑠𝑤

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interface can couple with each other and form a waveguide mode. For the waveguide with type II HMM as core and PR as claddings, the solution to TM0 mode is 𝛾0𝑑 = 2 arctanh(− 𝛼0𝜖𝑥𝑥⁄𝛾0𝜖1),

where the decay wavevector is 𝛼₀ = √𝛽₀2− 𝜖₁𝑘₀2 in the cladding and 𝛾₀= √𝜖_𝑥𝑥𝛽₀2/𝜖_𝑧𝑧− 𝜖_𝑥𝑥𝑘₀2 in the core made of HMM, 𝛽₀ is the tangential wavevector and 𝑑 is the thickness of the waveguide. Although 𝜖_𝑥𝑥 → 0 for both type I and II ENZ HMM, the behaviors are completely different. Specifically, the magnetic field 𝐻_𝑦(𝑧) and 𝐻_𝑦(𝑥, 𝑧) in the Al based type II ENZ HMM are shown in Figure 40A. The field in core of the waveguide are uniform, resembling a mode propagating horizontally because 𝐻_𝑦(𝑧) = 𝐻₀cosh(𝛾₀𝑧) → 𝐻₀, which is independent of 𝑧 when 𝛾₀ → 0 . Since the waveguide mode is in the forbidden zone for both HMM and PR, it is a plasmonic mode resonating at the interface of HMM and PR. The uniform wave front observed in the HMM (Figure 40A) when treated as an effective medium corresponds to the coupling of LRSP of individual Al2O3-Al-Al2O3 waveguide in the actual multilayer structure.

Moreover, the loss of the TM0 in the type II ENZ HMM is also much lower. As a result, the

propagation length and decay length becomes also much longer. The reduced loss in this case, as compared with typical type II HMM based lithography, greatly improves the process by significantly reducing the PR exposure time. Furthermore, the high 𝑘 of mode improves the resolution of the lithography. When the cutoff frequency is in the evanescent region i.e. 𝑘cutoff >

𝑘0, much smaller features can be generated. The resolution limit can be expressed as a function of

the cutoff frequency ∆≈ 4𝜋 𝑘⁄ _cutoff ≈ 2𝜆 𝑛⁄ _eff, where 𝑛_eff is the effective mode index determined by 𝛽₀/𝑘₀. In addition, the type II ENZ HMM can also benefit from filter effects [34] as illustrated in Figure 40B. In the 𝑘 space, the diffraction orders of the grating with period of 700 nm are given by the red arrows. The OTF, i.e., |𝐻_𝑡⁄ |𝐻_𝑖 2 _{shows a narrow band transmission, where the 𝐻}

𝑡 and

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diffraction of the grating is selected with its diffracted wavevector coinciding with the resonant peak of the OTF, while the other orders are blocked and cannot reach PR layer. With these characteristics, type II ENZ HMM is an excellent candidate in nanolithography for subwavelength patterns with high field contrast and uniformity.

Figure 40 Surface wave in a waveguide made of ENZ HMM.

(A) Magnetic field distributions 𝐻_𝑦(𝑥, 𝑧) in the 𝑥𝑧 plane (color) and 𝐻_𝑦(𝑧) along the 𝑧 axis (black curve) of distribution of the mode in the waveguide with Al based type II ENZ metamaterial as core and PR as claddings. The thickness of core is 165 nm. (B) Transmitted intensity |𝐻_𝑡⁄ |𝐻_𝑖 2

of 7 layers of 6 nm Al and 47 nm Al2O3 with the TM light illumination at 405 nm wavelength,

where the 𝐻_𝑡 is the transmitted magnetic field and 𝐻_𝑖 is the incident magnetic field. The diffraction orders of the 700-nm grating are marked by the red arrows.

A scheme of the ENZ UV lithography is shown in Figure 41A. One-dimensional (1-D) periodic Al grating is used as the photomask, and the type II ENZ HMM is placed in contact with a PMMA spacer as an index-matching layer, followed by the PR layer to record the imaged patterns. Additional Al and Al2O3 layers are placed beneath the PR layer to enhance the field inside the PR

by reflection and cavity resonance. [21,34,94] The Al grating is used in the mask to increase the field contrast because of the light coupling, and similar results can be obtained experimentally by Cr mask with Al based ENZ HMM as well. [13] In addition, the system is placed on a flexible polymer substrate to ensure intimate contact between the HMM lens and the PR layer. [22,34] To make a robust design, we also studied the effects of the geometric parameters in the design

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including the thickness of each layer, duty cycle and period of the grating etc. via simulation. Using the optimized design, the normalized electric field intensity distribution |𝐸|2 of the latent image in the PR layer is shown in Figure 41B for one mask period, illuminated by TM polarized light with a wavelength of 405 nm. The Al grating has thickness of 25 nm, duty cycle of 75%, and period of 700 nm. The PMMA spacer has thickness of 50 nm, which is on top of the HMM made of 7 stacks of 6 nm Al and 47 nm Al2O3 layers. The PR has thickness of 100 nm, the bottom Al

layer has a thickness of 20 nm, and the two Al2O3 films have a thickness of 47 nm. Within one

period, six periodic patterns are created, because the 3rd diffraction of the grating was employed for exposure. [34] An additional reduction factor of two comes from the interference of two counter-propagating waveguide modes. Comparing the normalized field distribution, the field intensity in the PR is strong, and the aspect ratio of the patterns is high. Moreover, the additional Al and Al2O3 layers underneath the PR not only further improve the contrast and uniformity of the

patterns in the simulations, but also reduce the film roughness in the fabrication. The field contrast shown in the simulation can be calculated as (|𝐸𝑚𝑎𝑥|2 − |𝐸𝑚𝑖𝑛|2) (|𝐸⁄ 𝑚𝑎𝑥|2+ |𝐸𝑚𝑖𝑛|2) ~ 0.915,

which is sufficiently high for exposure and to obtain high contrast ratio in images. The average field distribution |𝐸|2 is around 0.225, which is near a quarter of the incident light set a 1V/m. The |𝐸|2 _{distribution along horizontal lines in the PR is illustrated in Figure 41C, where the amplitude}

of the periodic distributions at the top, middle and bottom positions are comparable, which means that the field distribution is uniform along the vertical position. It should be note that the field is several orders of magnitude stronger compared with other system using HMM, [13,21] which is due to the low loss of the specially designed waveguide mode supported by ENZ HMM in our system. Moreover, based on our simulations, the dependence of the parameters such as the thickness of each layer is acceptable for the experiments, with tolerance of a few nanometers,

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which indicates that our design is quite robust. Due to the low loss and high anisotropy of type II ENZ HMM, the patterns in the PR are much smaller than the photomask while maintaining strong field intensity, high aspect ratio, and good uniformity.

Figure 41 Lithography design using ENZ HMM.

(A) Schematic of the ENZ lithography design in Cartesian coordinates. (B) Normalized |𝐸|2

distribution in the 𝑥𝑧 plane with the wavelength illumination for 7 layers of 6 nm Al and 47 nm Al2O3 ENZ HMM and the white dashed line indicates the middle position in the PR. (C)

Normalized electric field distribution |𝐸|2 _{along horizontal dashed lines at the top, middle and}

bottom positions in the PR.