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Limitations on Length Scales for Electrostatically Induced Submicrometer Pillars and Holes Leonard F. Pease, III and William B. Russel* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08444-5263 Received June 10, 2003. In Final Form: October 4, 2003 Thin leaky and perfect dielectric films can be driven electrically to form well-ordered patterns, typically of pillar arrays. While the technique appears to promise nanometer scale features, this paper begins to examine some of the limitations. The process, sometimes referred to as lithographically induced selfassembly, begins by spin coating a polymer onto a silicon wafer generating an initially featureless film and then overlaying a mask, which may be patterned, leaving a small gap. This configuration is heated above the glass transition temperature of the polymer, upon which flow ensues with a characteristic wavelength set by a combination of electrical forces and surface tension. The authors recently examined the initial stages of this process under pattern-free masks by deriving a generalized linear stability analysis not restricted to the lubrication approximation (J. Chem. Phys. 2003, 118, 3790). Herein comparison of this model with experimental data from the literature finds good agreement over a wide range of conditions including applied voltages and oxide layers on the mask and substrate. A significant discrepancy at the highest fields may be due to dielectric breakdown, suggesting that the minimum feature size may be limited. Viscous effects may also limit the effectiveness of large decreases in surface tension or large increases in electric field, leading to lower limits for the feature size. Long-range ordering seems to decrease as surface tension decreases and the potential increases, indicating that smaller pillars come with decreased quality. In the absence of an electric field, consideration of the viscosity-dependent time scale suggests an explanation for the apparent conflict between observations by Scha¨ffer et al. (Nature 2000, 403, 874) and those of Chou and Zhuang (J. Vac. Sci. Technol., B 1999, 17, 3197).
Introduction In a process, sometimes referred to as lithographically induced self-assembly (LISA),1 electrical forces, viscous forces, and surface tension conspire to pattern thin films into well-ordered arrays with micrometer length scales. The process, first reported by Chou and Zhuang1 and concurrently investigated by Scha¨ffer et al.,2 begins by spin coating a polymer onto a substrate, typically of silicon or ITO,3 generating an initially featureless film (see Figure 1). A small gap filled with air, fluid, or another polymeric/ organic liquid is left below a second wafer called the mask, the bottom surface of which may be patterned in relief. The mask-to-substrate separation ranges from 100 to 2000 nm and the polymer film is typically 50-750 nm thick. This configuration is heated above the glass transition temperature of the polymer, triggering flow in response to destabilizing forces. Observation and analysis by Scha¨ffer et al.2 confirms that these forces are electrical in nature, since development occurred only under an external voltage and the pillar spacings decreased commensurate with applied voltage for their high molecular weight polymers. Chou and Zhuang1 observed array formation for relatively low molecular weight poly(methyl methacrylate) (PMMA) in the absence of an applied voltage but indicated that intrinsic electric fields, perhaps arising from localized charge or contact potentials, were responsible. Zhuang4 later confirmed that increasing external * To whom correspondence may be addressed. E-mail: wbrussel@ Princeton.EDU. (1) Chou, S. Y.; Zhuang, L. J. Vac. Sci. Technol., B 1999, 17, 3197. (2) Scha¨ffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Nature (London) 2000, 403, 874. (3) Lin, Z. Q.; Kerle, T.; Baker, S. M.; Hoagland, D. A.; Scha¨ffer, E.; Steiner, U.; Russell, T. P. J. Chem. Phys. 2001, 114, 2377. (4) Zhuang, L. Controlled Self-Assembly in Homopolymer and Diblock Copolymer. Dissertation; Princeton University, Princeton, 2002; Chapter 2.
Figure 1. The initially flat film (solid) develops characteristically spaced undulations (alternating dash), the maxima of which are preserved in the subsequent “cone-shaped spikes” (short dash) and final pillars (not shown). Electrical forces from the applied voltage, V, or contact potentials, χi, are mediated by the dielectric constants of the gap, g, and film, , and the conductivity, σ, of the polymer film. The competition between the electrical forces and the opposing surface tension, γ, and viscosity, µ, set the characteristic spacing and time scales of the instability. Where oxide is present the dielectric constants are s and m and the thicknesses are -Hs and Hm - H for the substrate and mask, respectively.
voltages decreases pillar spacings for the same system. In either case, if the mask is not patterned in relief, an imbalance between electrical forces and surface tension causes sufficiently long wavelength fluctuations to be unstable,5 as observed by Lin et al.6 Cone-shaped spikes,7 (5) Pease, L. F., III; Russel, W. B., J. Non-Newtonian Fluid Mech. 2002, 102, 233. (6) Lin, Z. Q.; Kerle, T.; Russell, T. P.; Scha¨ffer, E.; Steiner, U. Macromolecules 2002, 35, 3971.
10.1021/la035022o CCC: $27.50 © 2004 American Chemical Society Published on Web 12/20/2003
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similar to cusps observed in experiments by Taylor and McEwan,8,9 emerge preserving the maxima in the perturbed film profile. These spikes contact with and spread on the mask forming well-developed pillars, which remain after the configuration is cooled below the glass transition temperature. Because the initial instability seems to determine the location of the spikes or cusps and subsequently the final pillars, accurate prognostication of the natural spacing of the initial undulations should reasonably forecast the pillar-to-pillar spacing of the final array. Arrays typically form hexagonally with spacings ranging from 1 to 47 µm and diameters ranging from 0.6 to 23 µm as summarized in the Supporting Information. Experimentally several trends have emerged. First, a wide variety of polymers may be employed including PMMA, polystyrene (PS), brominated polystyrene (PSBr), polyisoprene (PI), and polymer blends,4 though patterns develop more slowly with the higher molecular weight polymers employed by Scha¨ffer et al.10 than the lower molecular weight polymers of Chou and Zhuang.1 Experiments have been performed at temperatures ranging from 120 to 170 °C with indications that even lower temperatures approaching the glass transition temperature may work.11 Russell and co-workers10 applied voltages ranging from 19 to 60 V with larger voltages resulting in smaller pillars as seen via direct comparison of experiments on PS at 30 and 50 V. The spacing was inversely proportional to the applied field, all else held equal. Zhuang4 subsequently applied an external electric field of -40 to 90 V and found the spacing of the array to decrease slightly but to be relatively insensitive to the electric field. Thick layers of thermally grown oxide were present to reduce leakage current (60 nm on the substrate and 245 nm on the mask compared to the 105 nm mask-substrate separation). As surface tension suppresses this electrically driven assembly process, Lin et al.3,6 filled the air gap with oligomeric dimethylsiloxane (ODMS), polymeric dimethylsiloxane (PDMS), and PMMA to reduce the tension from ∼30 mN/m to as low as 1.2 mN/m. This reduction in interfacial tension and the addition of dielectric media decreased the pillar spacing by about a factor of 2 and increased the formation rate in one case by a factor of 50. They also found that a film occupying a majority of the mask-substrate separation yielded an array of holes filled by the upper fluid. The hole-to-hole spacing approximated that expected for a pillar-to-pillar spacing under the same conditions, indicating an electrohydrodynamic origin. Chou and Deshpande12 have also examined liquid/liquid interfaces and found the quality of ordering to be temperature dependent under patterned masks. If the mask is patterned, lateral gradients in the electrical field arising near edges13 of the relief draw in polymer from which pillars emerge sequentially at the corners, around the perimeter, and then in the interior, as observed by Deshpande et al.7 In each case new pillars arise first as characteristically spaced undulations, then as “cone-shaped spikes”, and finally as regular cylinders,7 (7) Deshpande, P.; Sun, X. Y.; Chou, S. Y. Appl. Phys. Lett. 2001, 79, 1688. (8) Taylor, G. I.; McEwan, A. D. J. Fluid Mech. 1965, 22, 1. Taylor, G. I. In The Scientific Papers of Sir Geoffrey Ingram Taylor; Batchelor, G. K., Ed.; Oxford University Press: Cambridge, 1971; Vol. 4, p 43. (9) Melcher, J. R.; Taylor, G. I. Annu. Rev. Fluid Mech. 1969, 1, 111. (10) Scha¨ffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Europhys. Lett. 2001, 53, 518. (11) Personal communication with Paru Deshpande. (12) Deshpande, P.; Chou, S. Y. J. Vac. Sci. Technol., B 2001, 19, 2741. (13) He, L.; Ouyang, Q.; Russel, W. B.; Chou, S. Y. Unpublished manuscript.
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though the spacing can differ by 50% between the interior and exterior.1 Regular pillar arrays, arrays of holes,6 gratings and rosettes,2 and concentric rings14,15 can be formed in like manner. A highlight of the work by Scha¨ffer et al.2 was the replication of a grating in PSBr with a line width of 140 nm. While thorough modeling and evaluation of patterns formed under a mask patterned in relief is beyond the scope of this paper, these remarkable structures motivate our effort to provide a stepping stone toward eventual understanding of these more complex structures. This process is particularly remarkable in light of recent efforts to surmount economically the 100 nm barrier for conventional lithography and to apply its benefits to less conventional materials.16-19 Practical utilization will require well-defined arrays, reasonable growth rates, and submicrometer length scales. Applications may include MEMS and microfluidics devices, novel storage devices, and photonic band gap materials.14,20 Considering the potential applications, we seek a thorough understanding of how the various parameters influence the physics of the process and the means for reducing the pillar/hole spacings and diameters. To accomplish these goals, models have been proposed for the process. The initial model assumed the film to be a perfect dielectric and employed a linear stability analysis in the context of the lubrication approximation. Scha¨ffer et al.10 showed the results of this analysis to match the pillar-to-pillar spacings of their experiments within a factor of 21/2. This is a remarkable result considering that linear stability analyses neglect later stages of development following the initial perturbation. Clearly the characteristic spacings are set very early in the process. Lin et al.3 extended their approach to incorporate a viscous dielectric media in the gap with similar agreement. Zhuang4 presented a variation on this theme by including the effect of static free charge in the normal stress balance at the free interface. Where this static charge is negligible, his results reduce to those of Scha¨ffer et al.10 He et al.21,22 have simulated the nonlinear evolution of the film and reproduced the sequential growth at the corners, on the edges, and then in the middle under square or rectangular masks. Inspired by Chou’s proposition1 that free charge drives the instability, we5 applied the leaky dielectric model under pattern-free masks to gain an understanding of the effects of conduction and accumulation of mobile free charge. As the presence and motion of free charge may be important, we employed the formulation originally put forth by Taylor and Melcher8,9 and clarified by Saville.23 The perfect dielectric used by previous investigators is a special case of this more general leaky dielectric when free charge is absent at the polymer-gap fluid interface. A linear stability analysis under the lubrication approximation suggested that the presence and motion of free charge can increase the rate of formation and decrease the (14) Chou, S. Y. MRS Bull. 2001, 26, 512. (15) Deshpande, P.; Pease, L. F., III; Chen, L.; Chou, S. Y.; Russel, W. B. Submitted. (16) Park, M.; Chaikin, P. M.; Register, R. A.; Adamson, D. H. Appl. Phys. Lett. 2001, 79, 257. (17) Wilder, K.; Quate, C. F.; Adderton, D.; Bernstein, R.; Elings, V. Appl. Phys. Lett. 1998, 73, 2527. (18) Noy, A.; Miller, A. E.; Klare, J. E.; Weeks, B. L.; Woods, B. W.; DeYoreo, J. J. Nano Lett. 2002, 2, 109. (19) Muther, T.; Schulze, T.; Jurgens, D.; Oberthaler, M. K.; Mlynek, J. Microelectron. Eng. 2001, 57-58, 857. (20) Suo, Z.; Liang, J. Appl. Phys. Lett. 2001, 78, 3971. (21) He, L. Unpublished manuscript. (22) He, L.; Ouyang, Q.; Russel, W. B.; Chou, S. Y. Unpublished manuscript. (23) Saville, D. A. Annu. Rev. Fluid Mech. 1997, 29, 27.
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spacing; the leaky dielectric yields both reduced spacings and faster growth rates than the perfect dielectric, as free charge cancels the field in the polymer and increases the force on the polymer-air interface. Subsequently, we24 presented a general linear stability analysis for both perfect and leaky dielectric films, which demonstrated the lubrication approximation to be least appropriate for the smallest pillars. This led to a discussion of how to tune the parameters developed in the model to achieve arrays with the smallest periods, highest aspect ratios, and largest growth exponents. Here we present a detailed comparison of our generalized model, which contains only measurable parameters, with data from the literature and discuss the implications regarding the minimum feature size, long-range order, and process time scales. Review of Model Results To the models derived previously, we add the effects of oxide/insulating layers on both the mask and substrate as recorded in the Supporting Information. We briefly summarize the oxide-free results and refer the reader elsewhere24 for further details. Figure 1 shows the relevant variables and geometry from which scaling identifies four independent dimensionless groups. Of the four, the most important group is a geometric parameter, H/L, where H is the mask-substrate separation and L represents the electrocapillary length set by competition between electrical forces and surface tension. The lubrication approximation, as previously mentioned, generally holds when the square of
H/L ) (goχ2/γH)1/2
(1)
is less than unity. Here g is the dielectric constant of the gap fluid, o is the permittivity of free space (8.854 × 10-12 C2/Nm2), γ is the surface tension, χ ) χ1 + χ2 + χ3 + V with χi as a contact potential,25,26 and V is the applied voltage on the backside of the substrate with the mask formally grounded. Clearly as the surface tension drops and the applied voltage rises, L decreases and H/L increases; either approach will lead to smaller pillars. Another important length scale ratio is the fill fraction
h h o ) ho/H.
(2)
As viscous forces scale on the initial film thickness of the polymer film, ho, and electrical forces scale on the thickness of the gap, H - ho, this ratio plays a major role in determining the relative balance. Two other dimensionless groups can be important in specific cases. When free charge is absent (i.e., the perfect dielectric) the dielectric contrast, /g, is important where is the dielectric constant of the polymer film. Increasing the dielectric contrast increases the growth rate and decreases the size of the pillar arrays. When mobile free charge is present (i.e., the leaky dielectric) the dielectric contrast and a conductivity parameter, S, play minor roles. The latter represents a ratio of the process time scale to the time for charge conduction (go/σ)
S ) σµγH 3/g3o3χ4
(3)
of the gap fluid. The process time scale,
H/U ) µγH 3/(goχ2)2,
(4)
signifies the motion of the interface with U as the characteristic vertical velocity and represents the competition between the electrical force that drives the process and the surface tension and viscosity that slow it. The first three dimensionless groups with their reported ranges are found in the Supporting Information. The largest value for h h o (0.72) resulted in an array of holes, so the largest value of h h o known to produce pillars is 0.53. Values for the bulk conductivity were not reported with the experimental data, but range over many orders of magnitude in the literature, from 10-12 S/m27 to 101 S/m28 (1 S/m ) 1 (C2 s)/(kg m3) ) 0.01 (Ω cm)-1), for a variety of polymers and concentrations of dopants. Thus, the conductivity parameter, S, may range from 10-1 to 1016 at room temperature where free charge is present. As the conductivities are known to increase by a couple of orders of magnitude when the temperature rises by 100-150 °C, we believe that S is typically much greater than unity, making charge conduction fast compared to motion of the interface. Several assumptions have gone into the two models, and we mention a few of them here. The mask and substrate are conductors (particularly true of the highly doped silicon used by Scha¨ffer et al.3 and perhaps less so for the lightly doped wafers of Chou and Zhuang1) and the insulating oxide layer inherent on any silicon wafer is of negligible thickness (an alternative derivation including these oxide layers is given in the Supporting Information). We also assume charge diffusion to be negligible. Free charge cannot transfer into the air or medium filling the gap and a net charge may accumulate at the boundaries, but not in the bulk. Thus, electrical stresses act only at the interfaces. The dielectrics are assumed to be linear and isotropic, and the fluid is essentially Newtonian. In the process of performing the linear stability analysis, we assumed that the initial film was perfectly flat and the charge (if present) was uniformly distributed, and we kept only terms first order in the perturbation. We assumed a scaled perturbation of the form h h ) h ho + ˜ is the amplitude of the h ˜ exp[i(kxx + kyy)], where h perturbation and kx and ky are the x and y components of the wavenumber. And finally we neglect any van der Waals forces. The growth exponent for the perfect dielectric, which depends on the wavenumber, k h , and on three parameters, H/L, h h o, and /g, is
h SoCo - k
m j ) 2k h3
(
H h h L o
H3 2 H2 2 k h h h o + Co2 3 2 L L
[(
H L To + g T1mo k h3
)
×
(
)
2 -1 g g h h o + g(1 - h h o)
)(
)
2
]
-k h 4 (5)
where σ is the conductivity and µ, the viscosity of the polymer, is assumed to be much larger than the viscosity
h ok h H/L), Co ) cosh(h h ok h H/L), To ) where So ) sinh(h tanh(h h ok h H/L) and T1mo ) tanh(k h H/L(1 - h h o)). The
(24) Pease, L. F., III; Russel, W. B. J. Chem. Phys. 2003, 118, 3790. (25) Sparnaay, M. J. The Electrical Double Layer; Oxford University Press: New York, 1972. (26) Akande, A. R.; Adedoyin, J. A. J. Electrost. 2001, 51-52, 105.
(27) Strzelec, K.; Tsukamoto, N.; Kook, H. J.; Sato, H. Polym. Int. 2001, 50, 1228. (28) Thakur, M. J. Macromol. Sci., Pure Appl. Chem. 2001, A38, 1337.
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leaky dielectric when charge conduction is fast, i.e., when S . 1, has a growth exponent
)[
]
H H h SoCo - k h h k h3 L o L 1 m j ) -k h4 . 3 2 2 T h o) 1mo 2 2 (1 - h 3 H 2 H 2k h k h h h o + Co L3 L2 (6)
(
The growth exponents when free charge is present but conduction is slow, i.e., when S is on the same order or less than unity, are not presented here as eqs 5 and 6 provide bounds. These low conductivity expressions can be found elsewhere when the lubrication is24 and is not5 relaxed. Also, conductivities were not reported with the data in the literature, so exact comparison is not possible. Under the lubrication approximation these expressions simplify considerably. For the perfect dielectric
(
) )
(
)
2 -1 g g 1 k h max ) 2 h h o + g(1 - h h o)
and
h h o3 m j max )
[
( )( g
2
-1 g
)
λmax ) tc )
]
)
(8)
3
) 6
]
(9)
1/2
3/2
g
|χ|( o) | - | 12µγ[gho + (H - ho)]6 χ4ho3(go)2( - g)4
(10)
(11)
These spacings are smaller than those of Lin et al.6 by a factor of 21/2. The predictions of Zhuang,4 where the effect of a static free charge, qs, was included in only the normal stress balance at the free interface, also reduce to this form with χ redefined as
χ ) V + χ1 + χ2 + χ3 + qsH/o(g - ), The expression for the growth exponent of the leaky dielectric with S . 1 also simplifies to
m j )
(
are simple functions of the initial height ratio as might be expected of a conducting film. The dimensional periodicity and the characteristic time are
23/2πγ1/2(H - ho)3/2
λmax )
(15)
|χ|(go)1/2
H/U 12µγ(H - ho) ) . m j max χ4h 3(g )2 o
(16)
o
()
H H , L L
|(
|
)
+ g gho + (H - ho) ) - g transition H
1/2
(17)
for the perfect dielectric or
()
H H , L L
)
transition
(
)
H - ho H
1/2
(18)
for the leaky dielectric. Transcendental expressions as in j max, eqs 5 and 6 must be solved numerically for k h max and m and the period, λ, is related to the wavenumber by
1 3 h h k h 4. 3 o max
2(2 )πγ [ ho + (H - ho)]
H/U ) m j max
(14)
Formally the lubrication approximation is satisfied when (H/L)2 , 1; however, in practice it suffices when
1/2
g
g
h h o3 1 1 3 ho k ) h h max4, 6 12 (1 - h 3 h o)
(7)
As these limiting forms under the lubrication approximation are sufficiently simple, unscaled analytical expressions for the predicted periodicity and time scales can be expressed as 1/2
m j max )
4
12 h h o + g(1 - h h o)
1/2
(13)
and
tc )
The two terms on the right represent the electrical and surface tension contributions, respectively. Clearly there is a maximum in a plot of m j versus k h as these two competing terms differ in sign. The wavenumber and growth exponent of this maximum are given by
([
k h max ) 2-1/2(1 - h h o)-3/2
6
2 1 3 h h -1 3 o g g 1 3 4 h k k h2 - h h . m j ) 3 3 o h h o + g(1 - h h o)
(
The maximum wavenumber and growth exponent,
)
1 3 2 1 h h k h -k h2 . 3 3 o (1 - h h o)
(12)
λ)
2πH 2πL ) k h max (H/L)k h
(19)
max
for square packings. Alternatively, the packing may be hexagonal as described by Taylor8 with a Christopherson distribution, modified here as
˜ {exp[ik(31/2x/2 + y/2)] + h h)h ho + h exp[ik(31/2x/2 - y/2)] + exp[iky]}. This perturbation is indistinguishable from the square packing under linear theories,8 but the corresponding spacing is found by replacing the 2π in eqs 10, 15, and 19 with 4π/(31/2). Thus, the hexagonal packing densities are 86% of the square densities. In either case, the spacing should decrease as surface tension decreases and as the applied field increases, since the spacing is an explicit function of H/L. Note that the electrocapillary length scale alone almost always underestimates the pillar/hole spacing as k h max is usually less than 2π (or 4π/(31/2)). If we assume that the final pillar packing is square planar, the fluid incompressible, the pillar walls perfect cylinders, and the final pillar-to-pillar spacing given exactly by the maximum wavelength predicted by the model, then the pillars have an aspect ratio of
( )
H 1 H )k h max D L 16πh ho
1/2
(20)
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Figure 2. The spacing to mask-substrate separation ratio, λ/H, versus the initial height ratio, h h o, for the perfect dielectric (dash) and leaky dielectric with S . 1 (solid) color coded for each set of data. Open diamonds are for PS/air at 30 V10 (orange), 50 V10 (royal blue), 30 V6 (red), and 50 V3 (green). Open squares are for PSBr/air at 30 V10 (purple) and a variety of voltages (pink).5 Open triangles are for PMMA/air at 30 V10 (green) and at 37 V AC10 (brown). Open circles for PI/air at 20 V3 (pink).
where D is the diameter. A similar balance for holes yields
(
H H 1 )k h max D L 16π(1 - h h o)
)
1/2
.
(21)
In the case of the hexagonal perturbation and packing, the 16 becomes a 24/(31/2) in both formulas. The difference is, however, minor and not meritorious of emphasis. We now proceed to compare these model predictions to data for the spacings, the aspect ratios, and time scales for growth. Results and Discussion The discussion to follow finds the data available from the literature to be consistent with the foregoing models and identifies key areas in need of further experimentation. Dielectric breakdown and viscous effects are highlighted as phenomena that limit reduction of the periods and diameters into the submicrometer length scales. We conclude with the implications of the theory regarding ordering, aspect ratios, and characteristic times. Spacings. Figure 2 displays much of the data available from the literature. Of the four dimensionless parameters on which λ/H may depend, the initial height ratio or fill fraction, ho/H, and the dielectric contrast, /g, are reasonably well known, though in certain cases the latter has to be estimated or the temperature of the reported value is not clear. These cases are clearly delineated in the table in the Supporting Information with deviations of less than 30% due to thermal expansion. The other two parameters, H/L and S, are somewhat more tenuous as they require the contact potentials and the conductivity of the polymer, neither of which were reported. To compensate, we neglect the contact potentials where the applied potentials are nonzero and provide a best-fit estimate otherwise (see the Supporting Information). As discussed previously, we assume the conductivity parameter, S, to be much larger than unity and present the leaky dielectric accordingly. A vast majority of the data in Figure 2 is generally bounded by the lines for the perfect (dashed lines) and leaky dielectrics (S . 1, solid lines), showing that our predictions bracket the data. Not surprisingly much of the data correlate slightly better with the perfect dielectric
as demonstrated previously by Scha¨ffer et al.10 though we have included the factor of 21/2 previously neglected in the electrostatically induced pressure. A few of the data sets, particularly the PSBr and perhaps some of the deuterated PS (dPS), correlate better with the leaky dielectric, suggesting the presence of free charge. Assuming hexagonal instead of square packing would move the lines for the theory up by a factor of 2/(31/2) ≈ 1.15, bringing the leaky dielectric (S . 1) into slightly better agreement at the expense of the perfect dielectric. However, the observations often did not report the type of packing (i.e., hexagonal or square), so this refinement was not attempted. Lack of indisputable correlation with either the perfect or leaky dielectric formulations suggests experiments monitoring the presence and conductivity of any free charge. In either case, the agreement supports the contention of Scha¨ffer et al.10 that linear stability analyses are generally sufficient to predict the final spacings. The predictions in Figure 2 do not invoke the lubrication approximation, but the difference for the polymer/air data is at most 15%, as these values of H/L generally do not exceed unity. Figure 3 shows data for which the gap was filled with ODMS, PDMS, or PMMA (filled symbols), which reduce the interfacial tension compared to their counterparts with air-filled gaps (open symbols in Figure 2) by about an order of magnitude. These lower interfacial tensions give rise to smaller electrocapillary lengths, L, and larger values of H/L, which should result in smaller spacings. Nevertheless, the decrease in spacing is less than the decrease in L that accompanies the insertion of a dielectric media in the gap, due to the decreased dielectric contrast for the perfect dielectrics. In contrast, lowering the interfacial tension greatly enhances the spacing for the leaky dielectrics as the electrostatic force is preserved. This would result in a full order of magnitude drop in the spacing, simply by changing the free charge content while holding constant all other the thicknesses and material properties. Lubrication Approximation. Figure 4 shows the effect of lateral electrical and viscous forces neglected under the lubrication approximation, which requires both the thickness of the film and the thickness of the gap be
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Figure 3. The spacing to mask-substrate separation ratio, λ/H, versus the initial height ratio, h h o, for the perfect dielectric (dash) and leaky dielectric with S . 1 (solid) color coded for each set of data. Blue circle for PI/ODMS at 20 V.3 Brown diamond for PS/ODMS 3 6 6 at 50 V. Filled diamonds are for PS/PDMS at 50 V (pink) and at 50 V (orange). Also shown is PS/PMMA at 30 V6 with a green diamond and PMMA/PDMS a variety of voltages6 with purple triangles.
Figure 4. The ratio of the actual spacing, λ, to the spacing under the lubrication approximation, λlubrication, versus H/L for the perfect dielectric with /g ) 10 for ho/H ) 0.1-0.9. Electrical effects lower λ relative to λlubrication whereas viscous effects increase λ.
less than the peak-to-peak spacing of the instability, i.e., ho/λ and (H - ho)/λ must both be less than unity. On the basis of eq 19, these may be reexpressed as
ho ho k h max H ) λ H 2π L
(22)
H - h o H - ho k h max H ) λ H 2π L
(23)
Figure 5. The spacing to mask-substrate separation ratio, λ/H, versus H/L for the perfect dielectric with /g ) 10. The lubrication approximation predicts a slope of -1. Slopes steeper than -1 derive from lateral electrical forces, and slopes less steep than -1 arise from viscous forces. The values of (H/L)transition for each of the initial height ratios, ho/H, are marked with a cross.
and
showing that we should anticipate departures from the lubrication approximation when H/L exceeds unity. When (H - ho)/H ≈ 1 and H/L . 1, the lateral electrical forces dominate and drop λ/λlubrication below unity in Figure 4. But when ho/H ≈ 1 and H/L . 1, the lateral viscous forces are no longer negligible, causing λ/λlubrication to increase. In all cases, however, sufficiently large H/L produces lateral viscous forces that counter any increases in the lateral electrical forces, eventually causing all curves to increase above unity. The effect of these two forces on the spacing is seen in Figure 5. Under the lubrication approximation a plot of λ/H versus H/L has a slope of -1. Lateral electrical forces
Figure 6. The spacing to mask-substrate separation ratio, λ/H, versus initial film thickness ratio, ho/H, with H/L ) 10000 for the perfect dielectric with /g ) 0.5 (short dash), 2 (long dash), and 10 (alternating dash) and the leaky dielectric with S . 1 (solid).
increase the magnitude of this slope while viscous forces decrease it. Spacings continue to decrease as H/L increases with each succeeding decade, albeit more slowly as viscous effects become more dominant. Figure 6 shows the theoretical predictions of λ/H for H/L ) 104, which is orders of magnitude beyond that achievable experimentally with current techniques. For h h o < 0.6, λ/H falls below unity
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Figure 7. log10(λ (µm)) versus log10(H - ho (nm)), log10(gho + (H - ho) (nm)), and log10(gho + (H - ho) + g(Hm - H)/ m (nm)) for the leaky dielectric S .1 (solid and diamonds), perfect dielectric (short dash and squares), and perfect dielectric with insulation (alternating dash and circles), respectively. A single set of data4 is plotted with the three different abscissi with s ) m ) 4.0. Table 1. Table of H/L Values for Filled-Gap Experiments polymer (kg/mol)
gap (kg/mol)
PI (40)a PS (0.58)a PS (30.3)b PS (30.3)b PS (30.3)b PS (30.3)b PS (96)b PMMA (27)b PMMA (27)b PMMA (27)b PMMA (27)b
ODMS ODMS PDMS PDMS PDMS PDMS PMMA (27) PDMS PDMS PDMS PDMS
a
(H/L)transition (H/L)transition H/L perfect leaky experiment dielectric dielectric 1.73 3.14 2.82 2.67 2.95 3.56 5.86 2.04 1.37 1.80 3.43
9.67 12.47 10.54 10.59 10.72 10.60 4.44 2.61 2.62 2.57 2.65
0.93 0.93 0.71 0.75 0.84 0.75 0.53 0.89 0.89 0.86 0.92
Data from ref 3. b Data from ref 6.
and below 0.1 for h h o < 0.2. Similar trends are found for H/L ) 10 though the λ/H curves are higher by about an order of magnitude. The implication is that even for the small values of γ and the large values of χ necessary for H/L ∼ 10, it will be challenging to lower λ much below H and even more difficult to lower λ by an order of magnitude. Thus, the benefit to be gained by decreasing surface/ interfacial tension is limited, in addition to well-known difficulties with interfacial thickness.29 Also in Figure 5 crosses designate the value of (H/L)transition as described in eq 17 by which deviations from the lubrication approximation begin. Thus, one should employ the lubrication approximation only when H/L , (H/L)transition. For the data shown in Figure 3, the values of H/L were less than (H/L)transition for the perfect dielectric as seen in Table 1. The sole exceptions are the data for PS/PMMA and PMMA/PDMS on the last line of the table where H/L exceeds (H/L)transition, but in these cases the differences between λ with and without lubrication are merely 14 and 36%. The values of H/L, however, do exceed (H/L)transition for the leaky dielectric, again indicative of the advantages of free charge in the polymer film. Observation of significant deviations from the lubrication approximation and the accordingly smaller (29) Merfeld, G. D.; Karim, A.; Majumdar, B.; Satija, S. K.; Paul, D. R. J. Polym. Sci., Part B: Polym. Phys. 1998, 36, 3115.
spacings predicted by the model would necessitate further experimentation. Dielectric Breakdown. Figure 7 shows the importance of accounting for insulating layers where present. Rearranging (10) and (15), and similar forms including oxide layers (found in the Supporting Information), finds
log(λ) )
3 log[gho + (H - ho)] + 2 log
for the perfect dielectric and
log(λ) )
(
)
23/2πγ1/2 (24) |χ|(go)1/2| - g|
(
)
23/2πγ1/2 3 log(H - ho) + log 2 |χ|(go)1/2
(25)
for the leaky dielectric without oxide layers. The slope of a plot of log(λ) against the logarithm containing H and ho should be 3/2 and independent of the (unknown) contact potential. Data by Zhuang4 for which the oxide thickness, initial film thickness, and mask-substrate separation were systematically varied, yield a slope of about 4/5 (i.e., the diamonds and squares) when plotted according to eqs 24 and 25, which fail to account for the oxide layers. When the oxide is incorporated, the fit slope increases significantly, exceeding the 3/2 prediction. The slope would come closer to 3/2 if the oxide were slightly thinner. Thus, existing oxide layers should not be neglected in the calculations. Another constraint on the process is illustrated in Figure 8, which depicts the spacing versus voltage as reported by Zhuang.4 When the sum of the applied voltage and the contact potentials approaches zero, the electrocapillary length diverges as the electrical force becomes negligible compared to surface tension. The predictions (solid and dashed lines) are symmetric about χ ) 0 and vary approximately inversely with the applied voltage. The data similarly rise in the middle of the plot, corresponding to a sum of contact potentials of approximately -30 V. While these contact potentials appear inordinately large, some reports26,30 have indicated contact potentials for polymers
802
Langmuir, Vol. 20, No. 3, 2004
Pease and Russel
Figure 8. Spacing, λ (µm), versus applied voltage, V (V) for data (circles)4 compared with perfect dielectric (long dash), leaky dielectric (solid), and perfect dielectric with oxide layer (short dash) predictions where ) 3.6, s ) m ) 4.0, γ ) 32.7 mN/m and χ ) V - 30 V. Table 2. Table of Electric Fields in the Polymer Film and Air Gap applied voltage (V)
gap field leaky dielectrica (MV/m)
gap field perfect dielectrica (MV/m)
PMMA field perfect dielectrica (MV/m)
-40 -11 0 20 25 39 60 90
-554 -325 -238 -79 -40 79 238 475
-495 -290 -212 -71 -35 71 212 424
-137 -80 -59 -20 -10 20 59 118
a
Estimates incorporate the -30 V offset with s ) m ) 4.0.
of a few volts. Furthermore, the presence of fixed charge at one or the other of the interfaces could significantly increase the apparent contact potential. Zhuang included a static charge at the interface to account for the 30 V offset in his analysis under the lubrication approximation but neglected the corresponding lateral stresses. Charge present in the oxide layers of the mask or substrate or low doping might also contribute to these large apparent contact potentials. The discrepancy between theory and experiment away from +30 V is somewhat more perplexing. Relaxing the lubrication approximate does not account for this insensitivity to applied voltage though H/L exceeds O(1). Accounting for the rather thick 60 nm of oxide on the substrate and 245 nm of oxide on the mask (short dash for the perfect dielectric), incorporated to reduce the leakage currents present in many experiments,4,3 fails to resolve the discrepancy and also compromises the agreement near 30 V. The most promising explanation is dielectric breakdown in the gap. The fields in the gap and in the PMMA film shown in Table 2 are several hundred megavolts per meter at the voltage extremes. Given the relatively small gaps and the large applied voltages, we suspect Fowler-Nordheim emission,31 which historically has been observed when gaps are less than a few (30) Wolfe, C. M.; Holonyak, N., Jr.; Stillman, G. E. Physical Properties of Semiconductors; Prentice Hall: Englewood Cliffs, NJ, 1989; p 329. (31) Fowler, R. H.; Nordheim, L. Proc. R. Soc. London, Ser. A 1928, 119, 173.
micrometers and fields are larger than about 80 MV/m.32-34 We speculate that this emission exchanges electrons with the polymer interface, shielding the interface from the full effects of the applied voltage and, thus, producing the unexpectedly large spacings. Additionally, Fowler-Nordheim emission is qualitatively consistent with the small, but undesirable, leakage currents reported. For the polymer films breakdown is probably not a concern as estimates of the required fields are at least a factor of 2 larger35 than those shown in Table 2. Thus dielectric breakdown represents a second limitation on reduction of the pillar/hole spacing. Whereas, the theory indicates that the largest reasonable fields are necessary to decrease the length scales, prevention of breakdown requires appropriate selection of masksubstrate separations and applied voltages. These two conditions bound the choice of the applied potential. If we rearrange eqs 17 and 18 in terms of χ (i.e., rearrange H/L > (H/L)transition) so as to capitalize on the lateral electrical forces and select χ so as to not exceed the field strength of the gap, Ebgap, then an inequality can be constructed which bounds the value of χ for the leaky dielectric
(
)
γ(H - ho) go
1/2
< |χ| < (H - ho)Ebgap
(26)
and the perfect dielectric
|
|(
)
gho + (H - ho) + g γ - g go
1/2
< |χ|
