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Fabrication of 3D fractal structures using nanoscale anisotropic etching of single crystalline
silicon
View the table of contents for this issue, or go to the journal homepage for more 2013 J. Micromech. Microeng. 23 055024
(http://iopscience.iop.org/0960-1317/23/5/055024)
IOP PUBLISHING JOURNAL OFMICROMECHANICS ANDMICROENGINEERING
J. Micromech. Microeng. 23 (2013) 055024 (10pp) doi:10.1088/0960-1317/23/5/055024
Fabrication of 3D fractal structures using
nanoscale anisotropic etching of single
crystalline silicon
Erwin J W Berenschot, Henri V Jansen and Niels R Tas
MESA+ Institute for Nanotechnology, University of Twente Enschede, 7500AE, The Netherlands E-mail:H.V.Jansen@utwente.nl
Received 7 February 2013, in final form 22 March 2013 Published 18 April 2013
Online atstacks.iop.org/JMM/23/055024
Abstract
When it comes to high-performance filtration, separation, sunlight collection, surface charge storage or catalysis, the effective surface area is what counts. Highly regular fractal structures seem to be the perfect candidates, but manufacturing can be quite cumbersome. Here it is shown—for the first time—that complex 3D fractals can be engineered using a recursive operation in conventional micromachining of single crystalline silicon. The procedure uses the built-in capability of the crystal lattice to form self-similar octahedral structures with minimal interference of the constructor. The silicon fractal can be used directly or as a mold to transfer the shape into another material. Moreover, they can be dense, porous, or like a wireframe. We demonstrate, after four levels of processing, that the initial number of octahedral structures is increased by a factor of 625. Meanwhile the size decreases 16 times down to 300 nm. At any level, pores of less than 100 nm can be fabricated at the octahedral vertices of the fractal. The presented technique supports the design of fractals with Hausdorff dimension D free of choice and up to D= 2.322.
(Some figures may appear in colour only in the online journal)
1. Introduction
The word fractal is Latin for ‘fractured’ and is used in disciplines dealing with broken geometries. In such geometries—after magnification—the shape appears identical, so, the magnified piece is an (almost) exact copy of the whole [1–5]. The fascinating fact about fractals is the variety of their appearance and applications. Besides in beautiful mathematical expressions [6–10], they show up in almost every observable identity in nature; from particles to landscapes [11,12], from earth to galaxies [13–18], from molecules via protein and DNA to life [19–28], even illness and—finally— death [29–32]. Also, fractal theory has been used for fractured material [33,34], fluids [35–39], physics [40–45], chemistry [46–49], electronics [50,51] and graphics [52–55].
Although the appreciation of fractals can be traced back to BC, in mostly religious architecture and artwork, one of the earliest mathematical descriptions was introduced by von Koch in 1904 [56], when he showed how a triangle transforms into a snowflake after a few recursive operations as demonstrated
in figure1. However, the subject of fractals gained popularity only after the recognition by Mandelbrot that many ‘chaotic’ phenomena in nature can be described in a comprehensive way using mathematical functions. In 1967 he questioned the length of the coastline of Great Britain being infinite or not [57], and in 1975 Mandelbrot published his fractal ideas in ‘Les objets fractals, forme, hasard et dimension’ [58].
Despite the enormous popularity and interest of fractals in scientific literature, most studies deal with the use of fractal theory to describe (or understand) various chaotic natural phenomena. Only a very few studies are concerned with the fabrication of fractals, such as fractal antennas [40], photonic crystal waveguides [59, 60], mixers [61], injectors [62], or heat transfer channels [32,63–65]. Consequently, the three-dimensional (3D) sculpturing of material using the recursive operation is a rather bare field. Of course, the fabrication of a fractal structure can be quite cumbersome, when all the parts have to be designed or created one by one [66]. It is therefore the aim of the current study to show that 3D fractals, as shown in figure2, can be engineered in a robust
Figure 1. Artificial 2D fractal: von Koch snowflake.
and time-efficient way. The technique uses for the first time the crystalline property of silicon to replicate small hollow octahedral features at the corners of bigger ones into a self-similar tree of ever smaller caves inside the crystal lattice. So, just like the von Koch snowflake [56], the crystal itself defines how the fractal develops. The silicon fractal can be used directly or as a mold to transfer the shape into another material. Moreover, the technique is able to produce a large quantity of identical structures in a very economical way due to the multiplication ability. In this paper, the fabrication sequence will be described and demonstrated with a few typical examples and some potential applications will be formulated.
2. Fabrication process flow
The engineering of the nanostructured fractals is based on a combination of anisotropic etching of silicon and corner lithography [67–70]. Corner lithography, as described in this study, uses conformal deposition and partly back-etching of a silicon nitride layer inside octahedral silicon pits to form wires or dots in sharp concave corners. The octahedral pits are a consequence of the crystalline property of silicon and the slow etching property of its1 1 1 planes during anisotropic wet etching [71, 72]. Corner lithography is a self-aligned technique as is edge lithography [73–78], but additionally
incorporates the ability to form truly 3D features. Fractals emerging from repeated edge lithography can multiply and scale down features too, but in essence stay planar [79]. By contrast, repeated corner lithography on top of anisotropic etching enables the construction of 3D solid octahedral fractals, fractals with holes and fractal networks of wire frames as will be demonstrated next.
The fractal process (figure3) starts with a1 0 0 single crystalline silicon wafer having a layer of thermally grown silicon dioxide (SiO2). The oxide is patterned in buffered
hydrofluoric acid (BHF) using a resist mask with a regular pattern of holes (6μm diameter and 12 μm periodicity). The unprotected silicon is anisotropically etched in a potassium hydroxide solution (KOH) to create pyramidal etch pits being ca. 4μm deep and the remaining oxide mask is stripped. The wafer is uniformly coated with 10 nm thin thermally grown oxide and 200 nm of low-pressure chemical vapor deposited (LPCVD) silicon nitride (figure3(A)).
The next step of the following iterative cycle is to partly remove the nitride in hot phosphoric acid (H3PO4),
first to leave nitride material in all the sharp concave edges (figure 3(B)), but finally only in the vertex corner of the pyramid as shown in figure 3(C); the process called corner lithography. Note that the thin oxide is used to protect the silicon from erosion by the H3PO4. Subsequently, the silicon
J. Micromech. Microeng. 23 (2013) 055024 E J W Berenschot et al
Figure 2. Engineered 3D fractal (bird view) using anisotropic silicon nanomachining. The lower image (top view) shows the ability to
machine such fractals’ wafer-scale.
is locally oxidized further into 80 nm SiO2using the nitride as
a mask; the so-called LOCal Oxidation of Silicon (LOCOS) [73]. The nitride in the pyramidal apex is stripped together with the thin oxide underneath. Then the unprotected silicon in the pyramidal vertex is etched anisotropically for 144 min, using tetra methyl ammonium hydroxide (TMAH). This forms a single octahedral shaped feature at the vertex of the pyramid
(figure 3(D)). Its size is determined by the 1 1 1 silicon etch rate (about 15–20 nm min−1). Then the LOCOS oxide is stripped, 10 nm thin oxide is grown and around 100 nm nitride is deposited. This finishes the zeroth level of processing.
The process from above is repeated to create the first generation or level of substructures: again, the nitride is partly etched (figure 3(E)), 80 nm LOCOS is grown, and 3
(A) (B ) (C) (D) (E ) (F ) (G ) (H ) (I ) (K ) (J )
Figure 3. Octahedral fractal fabrication scheme in the crystalline
silicon: start with surface initialization (A), the zeroth level (B)–(D) down to the fourth level (J, K).
the nitride and thin oxide are stripped in the vertices. Then silicon is etched, but using only half the etch time, i.e. 72 min TMAH. This creates five octahedra at the vertex corners of every octahedron of the previous step and half in size (figure 3(F)). The cycle is repeated for the second level to create 25 octahedra being again twice smaller as shown in figure 3(H). The procedure continues: the third level forms 125 octahedra (figure 3(I)) and the fourth level forms 625
octahedra (figure3(J)), etc. In order to visualize the octahedral fractal, the wafer is bonded upside down onto a borofloat glass wafer and the silicon wafer is sacrificed using a long TMAH etch, revealing the fractal structures bonded to the glass (figure3(K)).
3. Results
Figure4demonstrates with high-resolution scanning electron microscopy (HRSEM) images the effect of the iterative cycles on the silicon material. The image is a moment captured during the sacrificial silicon etch. Whereas the fractals at the left side have been totally liberated from the surrounding silicon mold material, the fractals at the right side are only partly released. In figure5(A), a close-up view of figure2, the cycle of the fourth level is interrupted after the nitride and thin oxide strip and the subsequent sacrificial silicon etch (this time the fractals are fully released). It shows a fabricated perforated silicon oxide fractal having pores at the vertices of the third-level octahedra with ribs of ca. 600 nm. The perforations are ca. 100 nm × 100 nm in size. Figure 5(B) shows the finished fourth level: after the thin oxide strip, additional octahedra are formed in TMAH and the interior is coated with nitride resulting in dense octahedra with ribs of ca. 300 nm.
Clearly, the sizes of the octahedral structures scale down, roughly with a factor 2 for every next level, while their number increases fivefold. The question rises how small the features can be and what the density might become. Evidently, the octahedral size of the next level should never pass half the size of the previous octahedron as this would cause the structure to merge during its expansion. In this perspective, one should realize that the connections between octahedra from a certain level and the previous level ask for some area. This will break the symmetry and limits the maximum allowed density. For instance having a closer look at figure 5(B), it is observed that the smallest octahedra at the vertices are indeed ‘separate units’, but the ocahedra along the ribs merge. In particular in the nanoregime, this issue becomes critical, so the etch rate should be monitored accurately and the aperture size should be minimized, when the maximum density is required.
The result from figure5can be slightly altered to construct another class of fractals: wireframes. Instead of 100 nm nitride, 1000 nm is deposited in the final process cycle. The reason for this is that the repetitive oxidation and stripping of the lower level octahedral features is diminishing the sharpness of those concave edges [76]. As a consequence of this rounding, the corner lithography is becoming ineffective and, therefore, a thicker nitride layer and more etch back is needed to form wires. In figure 6, the nitride etch back in the fifth level is intentionally limited to leave not only the nitride in the vertices, but also in all of the other sharp concave edges of the silicon mold (the ribs, like figure3(B)). As a result, a nitride fractal wireframe is fabricated. It is noted that due to the thicker nitride layer the octahedral features of the higher process levels are not wired, but solid octahedra are formed instead.
J. Micromech. Microeng. 23 (2013) 055024 E J W Berenschot et al
Figure 4. HRSEM image of fabricated perforated octahedral fractals. The fractals at the right side just emerge out of the silicon mold during
sacrificial etch.
4. Discussion
The essential feature of the fractal distribution is its scale invariance [4,5]; denote the number of details in the octahedral fractal of size si with Ni and the number of details of size
si+1with Ni+1. Then it follows:
log Ni+1 Ni = D log si si+1 , (1)
which when applied to the described octahedral fractal: every next level (i+1) provides the fractal with octahedral features five times more the previous level (i) and half the size and, thus,
D= log5/log2 = 2.322. The frequency–size distribution of the
octahedral structure obeys this relation, but only between the smallest and biggest octahedron and as such it is a nonrandom quasi-exact fractal (A mathematically exact fractal needs an infinite number of iterations to ensure that the detail is indeed an exact copy of the whole). As the Euclidean dimension of a surface is 2 and that of a volume is 3, we conclude that the octahedral surface has some volume aspects, which makes it an efficient surface for applications that interact with the surroundings via interfaces. Another interesting feature of the engineered fractal is its ability to have a fractal dimension free of choice below D = 2.322. For example, if the octahedral feature size of a next level would be scaled down by a factor of 5, the fractal dimension will become D= log5/log5 = 1. Furthermore, it is allowed to change the size of the next level at will and as such a nonlinear multifractal is formed. Despite the mathematical richness of these examples, we will return to the engineered fractal with a constant fractal dimension of
D= 2.322 and compute its relative surface area and apparent
porosity.
4.1. Apparent porosity
Assume that the process starts with a single pore in the apex of the parent pyramid. After one cycle this pore is replaced by five others and after i levels this will be 5i. We define the
apparent porosity as the single pore area times the pore density (the number of pores per occupied wafer area):
Pi=
total open area occupied wafer area = 5
i Apore
Apyramid
. (2)
For example, the fourth-level fractal of figure1carries 625 pores of 0.1μm × 0.1 μm per parent pyramid. The pyramid occupies an area of 12 μm × 12 μm, so P4 = 4%. It is
worthwhile to note that the apparent porosity can pass beyond 1 (i.e. 100%) after sufficient iterations (six in this case). This is caused by the fact that at higher fracturing levels, there are more than one fractal elements located at the same wafer spot, if only with a difference in wafer-depth position. Figures7(A) and (B) clarify this characteristic by showing the fourth-level octahedral fractal in orthogonal projection views. The black arrow is pointing at a xy-position where only a single pore is located. The gray arrow points at a spot with two pores at different depth (z-position) and the white arrow locates the xy-position where even four pores contribute to the perforation. This ‘super porosity’ with sub-100 nm pores is a notable result, e.g. because such a membrane would display efficient free molecular transport under ambient conditions [81].
(A)
(B)
Figure 5. HRSEM images of fabricated octahedral fractals. (A) Perforated oxide and (B) dense nitride.
4.2. Relative surface area
It is found for the effective area ratio (i.e. the total 3D surface of the fractal with respect to the occupied flat surface of the silicon wafer) that
Arel= 1.18 + 0.65 × 5× 5 4 i − 4 (3) in which the factor 1.18 stands for a constant offset due to the occupied flat silicon chip area including the pyramidal feature, 0.65 resembles the relative surface area of the
parent octahedron (i.e. scale factor a), and r = 5/4 is the common ratio. Clearly, this expression does not obey the linear characteristic of the ‘ordinary’ fractal (the log–log plot presented in figure 8is curved), although the added surface from every additional level is truly fractal. After performing four fractal sequences, the area is increased by a factor 6.5.
5. Potential applications
Although this paper does not concentrate on applications, but is merely a first demonstration of a new fabrication technology,
J. Micromech. Microeng. 23 (2013) 055024 E J W Berenschot et al
Figure 6. HRSEM images of silicon nitride wireframe fractals.
(A) (B )
Figure 7. Orthogonal projections of the fractal. (A) Top view and (B) cross-sectional view.
it is instructive to illustrate a few potential applications. Due to the huge possible apparent porosity it is evident that the perforated fractal membrane will find applications in the field of filtration and (free molecular) separation [80–85]. Likewise the extraordinary number of pores might be helpful as injector or spray nozzles and fast diffusion mixer devices. The property of a large effective surface area might be explored in capacitors, batteries, catalysis and dense membranes [86–91]. Furthermore, hollow and smooth fractal surfaces might be useful in trapping light for solar cell and photonic
applications [92–96]. Fundamental studies of light behavior when entering such a fractal structure will be very interesting to explore. The presented fractal wireframes might be selective particle traps [70], advanced 3D fractal antennas for enhanced device integration in RF applications [97,98], and coolers or heaters [99], which remove or distribute heat very uniformly and as catalytic frames. This can be useful in sensor applications or chemical reactors. Furthermore, instead of creating an additional level of octahedral structures, the perforated LOCOS oxide might be used to locally dope silicon 7
Figure 8. Relative area of a fractal as function of the number of
iterations.
and create a variety of 3D photonic crystals or quantum dot and wire arrays. As a last example, it is worthwhile to think about metallic or semiconducting dots or wireframes as potentially allowed in the suggested fractal fabrication scheme. This might be useful in the emerging field of 3D electronic integration.
6. Conclusions
In summary, a new class of 3D structures fabricated by a repeated silicon nanomachining procedure is demonstrated: octahedral fractals. It makes use of the crystalline property of silicon to form octahedral pits and corner lithography to define the location of the next level of octahedra in the silicon mold. Another essential feature of this technique is its ability to accurately form multiscale structures based on a single conventional lithographic step. We demonstrated the fabrication of dense, perforated and wire-like fractal structures. Starting from a single micron-sized mask opening, we have shown the creation of 625 nanopores for each individual octahedral fractal resulting in an apparent porosity of 4% and a fractal with a freely accessible area increased by a factor of 6.5 with respect to the occupied wafer area.
Acknowledgments
The authors appreciate the help of Mark Smithers in making the high-resolution SEM images.
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