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Thin metal films show a residual transmission for light in the visible and UV spectral range. This transmission can be strongly reduced by an appropriate sub-wavelength patterning of the metal film. Our investigation is focused on metal films with a thickness much below 100nm, where the transmission response is dominated by the individual posts acting like antennas and cannot be attributed to the excitation of surface plasmons. The almost complete suppression of transmission for ultra-thin metal films depends mainly on the absorber width, but not on the pitch of the pattern. The effect is robust with respect to imperfections of the geometry or larger features imprinted into the sub-wavelength pattern.
©2009 Optical Society of America
1. Introduction
The observation of enhanced transmission through an array of sub-wavelength holes or slits in a metal film by Ebbesen et al. [1] initiated many publications. The majority of these publications explains the mechanism with the excitation of surface plasmons (SP). The wave vector of surface plasmons in x-direction is given by [2]
where ε d and ε m specify the permittivity of the dielectric material and the metal, respectively. The symbol k = 2π/λ represents the wave vector of the incident light, λ is the wavelength in vacuum. There is still some controversy, whether surface plasmons really contribute to the transmission [3], and in which way [4]. A recent publication by Janssen et al. [5] reports that the transmission depends also on the phase of the surface plasmon. It is shown that an array of slits in a 200 nm thick gold film is nearly opaque, if the periodicity is an integer multiple of the SP wavelength.
The goal of this paper is to investigate transmission effects of sub-wavelength structures in extremely thin (<100nm) metal films. To facilitate an experimental verification, we focused our work on silver as absorber material and an illumination with a wavelength of 633 nm. Some of the simulation results which will be presented in the following sections cannot be attributed to the interference of surface plasmons. Instead of that, they can be understood in terms of an antenna model. We also investigate the sensitivity of the observed effects with respect to perturbations of the sub-wavelength periodicity by larger features (Section 4) and with respect to geometry variations such as oblique side-walls and corner-rounding (Section 5). The paper finishes with conclusions and an outlook.
2. Simulation setup
A schematic sketch of the investigated structures is shown in Fig. 1. The textured silver film with a thickness t and periodicity p is located on a glass substrate.
In the following, a 2D structure is specified by periodic silver ribs with a width m. As an example of a 3D structure we investigate quadratic posts with a mesa width m and an identical pitch p in both directions. In both cases the fill factor is given by f = m/p. The refractive indices were extracted from the RIT-page [6]. For λ = 633nm we have n glass = 1.4569 and n Ag = 0.136+i4.002. The pattern is illuminated from above along the z-axis by a TM polarized (i.e. x-direction) plane wave.
Simulations were performed with the Waveguide method [7, 8, 9, 10], also known as rigorous coupled-wave analysis (RCWA). The extracted 0th diffraction order of the transmitted (in glass) and reflected (in air) spectrum were normalized in such a way, that the normalized transmission T is equal to 1 for the bare glass substrate. A normalized reflectivity R = 1 corresponds to the reflection of a sufficiently thick film without a slit or mesa pattern.
Next, we introduce the relative transmission
to evaluate the impact of the sub-wavelength structure in comparison with the transmission of the unpatterned metal film T bulk. T rel varies between -1 and 1, and defines the optimum of T rel = 1.
3. Periodic slits and posts
First, we study a periodic array of slits. To employ the enhanced coupling to the metal via surface plasmons we focus on TM polarization (electric field perpendicular to the slits). Since we are interested in the thickness dependence, and as the transmission depends at least for thick layers on the pitch, we first investigate the relative transmission as a function of these two variables. Figure 2 shows simulation results for two different fill factors.
Fig. 2. Relative transmission T rel of a periodic array of slits in a silver film for different fill factors. The gray horizontal line corresponds to the wavelength in air (633 nm, dashed) and in the substrate (435 nm, solid). The curved black line specifies the thickness dependent surface plasmon wavelength [2].
For metal films with a thickness larger than 60nm a good agreement with the results from [5] exists. The transmission minima (or maxima of the relative transmission T rel) occur at pitches which correspond to integer multiples of the SP wavelength as determined by Eq. (1). In that case surface plasmons propagating along the air/metal and substrate/metal interfaces, respectively, are excited and energy is transferred to the metal film leading to a transmission minimum. The strongest transmission suppression results from the SP at the metal-substrate interface. Eq. (1) was derived for a semi-infinite intersection. Therefore, the observed deviation of the position of the transmission minimum for thinner metal layers is not surprising. The SP decay in positive and negative z-direction is given by the following formula [2]:
where d/m denotes the dielectric material or the metal, respectively. This results in a penetration depth of the SP in the silver film of about 24 nm. If the penetration depth of the SP becomes comparable to the thickness of the metal film, the interaction of SP on both sites of the metal film has to be taken into account leading to the curve displayed in Fig 2. It shows already a relatively good match with the position of the rigorously simulated transmission minima. Remaining deviations depend on the fill factor and can be attributed to the texture of the metal film. Note the linear correlation between pitch and thickness for very thin films with a thickness < 30nm.
Next we computed the relative transmission as a function of the fill factor and the pitch keeping the thickness constant. The results are shown in Fig. 3
Fig. 3. Relative transmission T rel of a periodic array of slits in a silver film for different thicknesses t. m indicates the constant ridge width of the hyperbolas, where the transmission is minimal. The gray horizontal lines are specified as in Fig. 2.
For very thin layers, the curve of minimum transmission in the pitch - fill factor plot has a hyperbolic shape. Hence, transmission minima occur for a constant ridge width. Also the wavelength dependence seems to be linear resulting in the following approximate expression for the position of the transmission minimum:
Such a dependence is typical for an antenna [11]. Hence, by reducing the film width the response of the system has turned from a more collective excitation of the whole grating to a merely individual response of quasi-isolated scatterers. The trenches act like dipoles, which are oriented perpendicular to the slits. In resonance these quasi antennas re-radiate the incident field with a π-shifted phase. The interference between the incident light and the antenna radiation cancels the light propagation in forward direction and re-directs the energy flux in the opposite direction. Additional simulations confirmed this general scheme for other substrate materials.
An alternative plot of the transmission canceling effect is shown in Fig. 4. In addition to the wavelength dependent transmission T, this figure shows also the normalized reflectivity R and the absorbed energy A within the metal layer. The thinner the films the more dominant is the resonant character of the response. The correlation between reduced transmission T and increased reflectivity R while the variation of the absorbed energy A is small confirms the proposed “antenna model”.
Fig. 4. Transmission T, reflection R and absorption A of an array of slits versus the wavelength for different values of the thickness t. The dashed lines show the values for metal layers without patterning. Pitch p and fill factor f correspond to the maximal relative transmission value according to Fig. 3.
Next, we will investigate whether this scheme still holds, for an arrays of 3D sub-wavelength patterns. In fact the transmission behaves similar to the slit case (see Fig. 5) thus we can adopt the explanation from above. Little posts act like antennas redirecting the energy flow when driven resonantly. In contrast to the 2D structures the optical response of the post array does not depend on the polarization of the incident light. Repeating the regression in the pitch independent region of reduced transmission again leads to an approximate formula for the position of transmission minima
The difference between the coefficients of Eq. 4 and Eq. 5 can be attributed to the fitting error.
4. Transmission effects in the vicinity of larger features
The proposed “antenna model” suggests, that the interaction between the individual ridges/mesas is less important for the observed decreased transmission values. The question arises, whether the features have to be periodic and whether the described effects can also be observed for more complex geometries. The first row of Fig. 6 shows top-views of the investigated geometries. Sub-wavelength arrays of mesas are placed around a 960 nm wide square opening. The number of sub-wavelength mesas placed around the opening increases from the left to the right. The pitch p = 240 nm and filling factor f = 0.8 of the sub-wavelength mesas represents a compromise between a good transmission supression (see Fig. 5) and manufacturability considerations. The center and lower row of Fig. 6 show the extracted field 1 nm and 100 nm below the metal film. Note the different scaling of the color bars.
Fig. 5. Relative transmission of an array of posts (right) in a 30 nm thin silver film. The horizontal lines are specified as in Fig. 2. The dashed hyperbola denotes the region of a constant ridge width m = 140 nm.
The sub-wavelength texture increases the electric field around the film significantly (center row 1 nm below the metal film). This field enhancement in the area of the sub-wavelength arrays can be attributed to a strong dipole which is excited inside the mesas. Due to the destructive interference of the dipole fields and the incident field, the transmitted light is almost completely canceled 100nm below the patterned metal film (lower row of Fig. 6). These results suggest that the sub-wavelength texturing of the metal film by few mesas suppresses the transmission of light in the vicinity of the contact hole. One potential application of the observed effect is the contrast enhancement for contact holes, which belong to the most critical features in semiconductor lithography.
5. Sensitivity with respect to geometry parameters
The proposed geometries have to be fabricated by a combination of lithography and dry etching processes. The limited resolution of the lithographic processes and the non-ideal pattern transfer in etching processes will result in variations of the fabricated sub-wavelength arrays from the ideal geometry as shown in Fig. 1. This section investigates the sensitivity of the observed transmission effects with respect to such geometry variations.
As can be seen in Fig. 5, the low transmission occurs for a relatively large range of pitches and fill factors. Ideal parameters for this 3D structure are p = 200 nm and m = 160 nm. According to our simulations, small deviations of the pitch (≈ ±15 nm), the fill factor (≈ ±0.08) and the thickness (≈ ±4 nm) are tolerable.
Figure 7 shows simulation results for other geometry variations such as corner rounding of the mesas/posts (left), oblique sidewalls of trenches (center), and incorrect etch-depths of the trenches. The variation of the ratio me/m (see inset in the left picture of Fig. 7) of an octagonal shaped post has only a weak impact on the resulting transmission. The case me = 1 corresponds to the ideal post. For me = 0 we get again a quadratic shape of the posts, with a pitch of 280 nm and f = 0.4. According to Fig. 5, such parameter combination still leads to a slightly reduced transmission. Additional simulations for circular posts (not shown here) confirmed this result. According to the simulation results which are presented in the center of Fig. 7, oblique sidewalls with an angle smaller than 20° have almost no impact on the transmission.
Fig. 6. Field distribution |E|2 of a contact hole (pitch 3840 nm, aperture width 960 nm) in a 30 nm thick silver film with different micro structuring. Polarization in y-direction. First row: structure geometry, second row: field 1 nm below the metal film (near field), third row: field 100 nm below the metal film (almost far field).
Fig. 7. Transmission (blue curve) of an array of posts (p = 200 nm, f = 0.8, t = 30 nm) with different topography variations as shown in the inset sketches. The dashed horizontal line depicts the bulk transmission of a 30nm silver film. The dotted red line in the right figure shows the transmission through a silver film with the thickness of the material in the trenches.
The right part of Fig. 7 shows the transmission of a 30nm thick metal film versus the cutting or etch depth. Incomplete etching of the trenches represented by a cutting depths below 30nm result in a strong increase of the transmission. A cutting depth smaller than 27 nm (more than 10 % material left) leads already to a value exceeding the bulk transmission. For a cutting depth of about 24 nm the film becomes almost transparent (T = 0.7). For cutting depths in the order of 20nm and below the transmission characteristics of the patterned metal film is simular to that one of a homogenuous metal film with the thickness of the remaining silver in the trenches. In contrast etching into the substrate (cutting depth > 30nm) has nearly no impact on the observed transmission values. Obviously the effect of reduced transmission vanishes only, if there is a shortcut between individual posts thus preventing the antenna action.
6. Conclusions and outlook
The transmission of thin metal films with appropriate sub-wavelength patterns can be considerably lower than that one of unpatterned metal films. If the thickness of the metal films is 100nm or larger, the observed effect is determined by the periodicity of the sub-wavelength array and can be attributed to the interference of surface plasmons. For very thin metal layers with a thickness of 40nm and below, the magnitude of the low transmission effect is governed by the size of the metallic features (ridges, mesas etc.). In that case, the observed effects can be qualitatively explained by a simple “antenna model”. The majority of the simulations for this paper were performed for (patterned) silver films and a wavelength of 633nm. However, additional simulations for other metals with a good conductivity, like gold, copper or aluminum showed similar results.
The observed effects occur also for small periodic sub-wavelength arrays in the vicinity of larger features like contact holes with diameters in the order of the wavelength or larger. Potentially this opens new possibilities for the contrast enhancement in contact hole imaging for lithographic applications. The observed effects are rather robust regarding small geometry variations such as corner rounding and oblique sidewalls. This alleviates the fabrication tolerances for an experimental verification of the observed effects.
Acknowledgments
The authors gratefully acknowledge funding of the Erlangen Graduate School in Advanced Optical Technologies (SAOT) and of the Cluster of Excellence Engineering of Advanced Materials - Hierarchical Structure Formation for Functional Devices by the German Research Foundation (DFG) in the framework of the excellence initiative.
References and links
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