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Abstract
A scheme to overcome diffraction limit in optical lithography via tunable plasmons is proposed. The plasmons are generated by a current-driven instability and are resonance amplified between the drain and source barriers of the transistor. A series of discrete deep subwavelength can be obtained by controlling the gate voltage. Thus, it is possible to realize lithography with a resolution over 1/100 vacuum wavelength and achieve arbitrary one-dimensional and even simple two-dimensional patterns. Our scheme works in the linear optics regime and is easy to be experimentally realized.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical lithography is widely used to print a circuit image onto a substrate [1]. However, according to the Rayleigh criterion, diffraction limits the minimum feature size to about half the optical wavelength [2]. To print smaller features, the illumination source should be reduced from visible light to extreme ultraviolet (EUV) or even X-ray. However, the air can absorb the EUV and X-ray significantly, and therefore it requires operation in a vacuum system. Moreover, the photon energies of the EUV and X-ray are so high that they may damage the silicon substrate [3]. Thus, finding a way to overcome this limit has been an attractive topic and many classical methods have been proposed to achieve superresolution in optical lithography [4–22]. However, the crucial parts of these proposals such as multi-photon absorption materials [4–7], multi-Λ levels [8,9] electronic structure, and strong laser with non-linearity [10–12] make them hard to realize in experiments. An alternative way is to shorten the wavelength via surface plasmons (SPs). SPs are electromagnetic excitations associated with charge density waves on the surface of a conductor or semiconductor. The wavelength of SPs can be much smaller than the vacuum wavelength, which inspires some proposals in lithography [18–22]. However, these schemes have a weak resolution and suffer from a fixed lithography pattern for determined plasmonic device [21].
In our study, a subwavelength lithography technique is proposed by using current-driven plasmons instabilities whose spatial frequency can be tuned by controlling the gate voltage [23,24]. As the resonance effect of the plasmons with the drain and source acting as cavity mirrors, a series of discrete deep subwavelength spatial frequencies can be obtained. In principle, arbitrary one dimensional and even two-dimensional targets with a resolution of up to two orders of magnitude beyond the diffraction limit can be achieved by the constructions of these spatial frequencies. Due to the linear nature of our scheme, little damage happens to the sample. Moreover, our scheme is easy to realize in experiments because it does not require any wavevector matching mechanism such as gratings [25,26].
The paper is organized as follows. In Sec. 2, we introduce our model. In Sec. 3, we present some numerical results and simulations. In Sec. 4, we present the concluding remarks.
2. 2DEG plasmonic dispersion relation
A schematic diagram of our system is shown in Fig. 1(a). A layer of two-dimensional electron gas (2DEG) acting as a transistor channel is formed at the interface of two semiconductor materials with slightly different band-gap energies. Plasmons are generated in the channel where the source and drain terminals are driven by a constant current source. Since both terminals are made of conductors, the plasmons form a standing wave in the channel region due to the reflection boundary condition. The structure is backed by a gate terminal of length L which controls the gate voltage and hence manipulates the carrier density in the channel. The gate terminal is either set to be far away from the channel or made of semiconductors in order to reduce the noise reflected from the gate terminal back to the channel. The thickness of the barrier layer is h. The molecular structure of the photoresist is shown in Fig. 1(b). Firstly, a laser beam in the visible frequency region with frequency ωe excites the molecules from the initial ground state |g〉 to the excited state |e〉. Then, SPs with frequency ωp excite the molecules to an ancillary state |a〉. Finally, a laser beam with frequency ωL dissociates the photoresist molecules in the state |a〉. Thus, the resulting patterns of the photoresist depend on the spatial distribution of population in the state |a〉.
Fig. 1 (a) Our scheme for lithography. The 2DEG is formed at the interface of the semiconductors where the standing plasmonic wave is generated by a current-driven instability. Transistor arrangement with S, D, G denotes source, drain, and gate, respectively. The surface carrier density is controlled by the gate voltage. (b) The energy structure of the photoresist.
The 2DEG can be modeled as a shunt admittance related to Drude-type surface conductivity [27]:
where Ns is the surface electron density in the channel, e is the electron charge, m* is the effective electron mass in the heterostructure, τ is the relaxation time of electrons, and ωp is the angular frequency of plasmons which is in resonance with the photoresist. The electron density NS can be controlled through the gate voltage Vg with the relation where N0 = ∊0∊2VT/ed is the zero-bias carrier density, VT is the gate threshold voltage of the transistor, ∊2 is the relative permittivity of the substrate semiconductor, and d is the distance between the 2DEG and the gate. The plasmonic dispersion relation is [25] where is the wave vector in dielectric i along z direction and k is the plasmonic wave number. In our scheme, ∊3 is the permittivity of air, and the permittivity of both semiconductor layers is approximated to the static value, i.e., ∊1 ≈ ∊2 = 9.5. Thus, the effective permittivity of the 2DEG is ∊(ωp) = ∊s − j∊s/(ωpΔ) [28], where Δ is the 2DEG thickness which is set to be 2nm in our simulations, and ∊s is the average permittivity of the 2DEG’s surrounding semiconductors. The photoresist transition frequency is ω0 = 2π × 25THz, the effective mass m* = 0.2me, the zero-bias density N0 = 7.5 × 1016m−2 and the scattering time is 1.14ps corresponding to a temperature of 77K [29]. As an example, we set the length of the 2DEG channel to be L = 2μm in our simulation. The photoresist is only resonant to the plasmons with frequency ωp = ω0. Therefore, we need to manipulate the wavelength of the plasmons with the frequency fixed at this value.The dispersion relation of SPs is shown in Fig. 2(a), where the plasmonic wave vector can be hundreds of times larger than that in vacuum. Figure 2(b) shows that the wave vector for a fixed frequency ωp = 2π × 25THz can be controlled through the carrier density in the channel which, according to Eq. (2), can in turn be manipulated by controlling the gate voltage Vg. Since the 2DEG channel behaves as a cavity due to the source and drain terminals acting as conducting boundaries, only wave vectors around (n is integer) can produce resonant sinusoidal patterns. The intensity distribution of the SPs at the bottom of the photoresist is studied with a full-wave simulation (COMSOL). The sinusoidal fringes with resonant(black) and non-resonant(red) wave vectors are shown in Fig. 3(a) where the resonant patterns bear less noise and waste less energy near the boundary and resemble sinusoidal patterns around the center. The resonant wave vectors can excite SPs with higher efficiency, but they only adopt certain discrete values.
Fig. 2 (a) The dispersion relation of SPs for our scheme. (b)The relation between the carrier density NS and the wavenumber k. Both (a) and (b) are the theoretical results when the size of semiconductor L is much larger than the plasmonic wavelength λ.
Fig. 3 (a) Fringes with resonant(black, NS = 6.67N0) and non-resonant(red, NS = 6.8N0) wavelength around 111nm. Intensity is normalized to 1. (b) Patterns with wavelength around 141nm. The input surface current is 3.36 × 10−2A/m. The red curve is of the form − cos(kx) with NS = 8.12N0, |A′| = 0, while the black curve is of the form cos(kx) with NS = 8.12N0, |A′| = 100V/m where U is the voltage between S and D.
3. Deep subwavelength lithography with 2DEG SPs
The following irregular one dimensional pattern of size 200nm is used to demonstrate that arbitrary pattern can be formed in our study(unit:nm):
The reflection boundary condition and the symmetry in our scheme result in the fact that the intensity of the standing wave in the channel is always symmetric and of the form ± cos(kx), and the sign depends on the value of k. For simple, we assume f(x) to be symmetric with respect to the center of the transistor and expressed as However, since the sign of an depends on the desired pattern f(x), the SPs may yield cosine fringes with phase differing from the desired Fourier component by π. As an example, for wavelength λ ≈ 141nm, since L/λ is an even number(≈ 14), the fringes must be of the form − cos(kx) which is shown as the red curve in Fig. 3(b). Thus, we need to shift the phase by π if we need cos(kx) pattern for λ = 141nm. In order to achieve this phase shift, we need to shine a plane wave with the same frequency normally onto the transistor with the polarization along x directrion. According to the simulation, Ex is of the form A + Ex cos(kx), and Ey is of the form Ey sin(kx) with Ex ≈ iEy. Thus the total field intensity isWhen the above external electromagnetic wave is applied, an additional constant is added to the x component of SPs, and the field intensity becomes
where A′ is a complex number depicting the magnitude and phase of the external field. Thus, when the phase and intensity of the plane wave are chosen properly, the fringes can be shifted by π, which is shown as the black curve in Fig. 3(b). In fact, by adjusting the plane wave, the fringes can be shifted with any phase, such as π/2 [25]. As a result, in principle, we can form arbitrary patterns with the above technique.Since the plasmonic frequency is in the Terahertz region while the laser ωe in Fig. 1(b) is in the visible light region, we can use the laser ωe to obtain the Fourier components with small spatial frequencies, which can be realized by counterpropagation of two beams with certain angles. For the lithography of the high spatial frequency components, we use plasmonic patterns as shown in Fig. 3 with certain wavelength by controlling the gate voltage.
Patterns of up to the 12th order (n = 12 and minimum wavelength λ = 167nm) for Eq. (5) are shown in Fig. 4. Resonant(non-resonant) sinusoidal fringes like the black(red) curves in Fig. 3(a)(Fig. 3(b)) are picked as the components for the black solid curve in Fig. 4(a)(Fig. 4(b)). As mentioned before, the resonant wave vectors do not necessarily match because of the non-perfect reflection boundary condition. Thus, the pattern with non-resonant wavelength as Fourier components matches the desired result with more accuracy. However, this comes with lower efficiency and only works in a small region near the center of the transistor because the plasmonic pattern beyond the region is not regular sinusoidal any more compared to the resonant pattern.
Fig. 4 The simulational pattern for the irregular one dimensional pattern Eq. (4) up to the 12th order is plotted as the black solid curve. The desired pattern according to the analytical result in Eq. (5) is depicted in the red dashed curve. (a) Components with resonant wavelength. (b) Components with non-resonant wavelength.
In a real lithography process, transistors are periodically located. As for our scheme, each transistor is 2μm wide and the width of the Drain and Source shown in Fig. 1 is 0.5μm. This means the transistors are periodically located with a period of 3μm. The etching areas are located in the middle of the transistors with the size of 1μm, which is controlled by the dissociation light ωL. The etching process for each transistor is the same as the above discussions, and the plasmonic field patterns on each transistor can be controlled separately and simultaneously. After the etching is finished by dissociating the photoresist with the dissociation light ωL, we move the target photoresist material to change its relative position to the periodically located transistors. Thus, since the period of transistors is 3μm, the whole area can be achieved by repeating the above process 3 times.
Arbitrary two-dimensional patterns can also be formed in principle by rotating the photoresist or along z axis. As a demonstration, considering the 2D pattern of the following form:
where A = {(x, y)|0 ≤ x, y ≤ 200}, B = {(x, y)|0 ≤ x, y ≤ 300}, C = {(x, y)|0 ≤ x, y ≤ 2000} (unit: nm). This pattern can also be expanded as Fourier series: Here we just keep the Fourier series up to the order where λ = 80nm. Since theoretically the wavelength of each term is not a fraction of the channel length L, it may not be possible to find resonant fringes with wave vector matching those terms in Eq. (9). However, in our scheme, the wavelength of each Fourier component can always match the closest resonant wavelength of SP with tolerable error. Therefore, arbitrary two-dimensional pattern can be in principle achieved. In Fig. 5, we simulate the two-dimensional pattern, which has a resolution up to 100nm. However, etching a two-dimensional pattern is much harder than a one-dimensional one because more components in different directions are needed. Thus, the background deposition is larger [10] and accumulative error is not negligible.4. Conclusion
In conclusion, we propose a scheme to realize subwavelength lighography via tunable SPs, which may be useful in nanodevice fabrication. The wavelength can be shortened to more than 1/100 vacuum wavelength with a resolution of tens of nanometers. Compared to the previous schemes, this scheme works in the linear optics region and is based on transistor which is technically easy to realize in experiments. We demonstrate that arbitrary one-dimensional and even simple two-dimensional patterns can be realized with satisfactory accuracy.
Funding
King Abdulaziz City for Science and Technology(KACST); National Natural Science Foundation of China (11804219).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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