CHUCKING PRESSURES FOR IDEALIZED COULOMB-TYPE ELECTROSTATIC CHUCKS

Transcription

1 Report No. Structural Engineering UCB/SEMM-2011/04 Mechanics an Materials CHUCKING PRESSURES FOR IDEALIZED COULOMB-TYPE ELECTROSTATIC CHUCKS By Ger Branstetter Dr. Sanjay Govinjee June 2011 Department of Civil an Environmental Engineering University of California, Berkeley

2 CHUCKING PRESSURES FOR IDEALIZED COULOMB-TYPE ELECTROSTATIC CHUCKS G.BRANDSTETTER, S. GOVINDJEE JUNE 02, 2011 Coulomb-type electrostatic chucks are consiere to be a key-technology for the next generation extreme-ultra-violet lithography. The electrostatic pressure hols the photo-mask uring the fabrication process in vacuum. Different formulas appear in the literature on how to relate this electrostatic pressure to the applie voltage on the chuck electroe. We iscuss the physical meaning of the corresponing formulations an also consier the implications for correct bounary conitions uring finite element simulations. 1. Introuction In orer to moel the forces acting in a Coulomb-type electrostatic chuck system holing a photo-mask, assume an ieal capacitor as picture in Fig. 1. Two conucting plates are separate by the chuck ielectric with thickness an relative permittivity ǫ r, an a vacuum gap of size δ a > 0. Let the lower plate represent the chuck electroe which is fixe in space. The upper plate is assume to be a rigi mask with a conucting back-sie layer, which is hel by a force F to ensure static equilibrium. We assume the ielectric to be perfect in the sense that no leakage current occurs; i.e. we focus in this F x Mask back-sie p +δ a V Chuck ielectric ǫ r p Electroe p c 0 Figure 1: Schematic of electrostatic forces acting on mask back-sie an chuck ielectric/-electroe when δ a > 0. 1

3 2 V ǫ r p c,0 p c,0 x 0 Figure 2: Schematic of electrostatic forces acting on mask back-sie an chuck ielectric/-electroe uring full contact. analysis on a Coulomb-type electrostatic chuck as oppose to a Johnsen- Rahbek electrostatic chuck (see e.g. [1]). We also assume that the ielectric is rigi, homogeneous, linear an neutrally charge. We will show that ue to a potential ifference V between the conucting mask back-sie an the chuck electroe, the following pressures are acting on the mask back-sie, the top surface of the ielectric an the chuck electroe respectively: p(δ a ) = ǫ 0ǫ 2 rv 2 2(+ǫ r δ a ) 2 (1) p (δ a ) = ǫ 0(ǫ 2 r ǫ r )V 2 2(+ǫ r δ a ) 2 (2) p c (δ a ) = ǫ 0ǫ r V 2 2(+ǫ r δ a ) 2, (3) where ǫ 0 is the permittivity of free space an equal to C 2 /Nm 2. Recent literature commonly agrees on the use of (1) as the pressure acting on the mask back-sie (see e.g.[2, p.45], [3], [4], [1], [5], [6], [7] or [8]). An early publication by [9] states (3) as the electrostatic pressure acting on the mask an chuck. We fin this to be incomplete an require (2) acting on the ielectric, such that the overall electrostatic pressure acting on the chuck as up to (1). Consier next the situation where F = 0; i.e. the mask is in contact with the chuck ielectric as illustrate in Fig. 2. As we will show, the contact pressure acting at the top an bottom surface is equal to p c,0 = p c (0) = ǫ 0ǫ r V (4) This formula was establishe by [9] an is use for example by [10], [11], [12], [13], [14], [15], [16] an [5]. However, recent publications by [7] or [4] propose to use p(0) as the chucking pressure when δ a = 0, as oppose to p c,0 which iffers by a factor ǫ r. In the following iscussion we argue that both formulations have their valiity, epening on their use for example in finite element simulations.

4 2. Electrostatic Pressure Calculation Let us first erive the electrostatic pressures for the ifferent cases mentione above. We will carry out a irect calculation via the Maxwell stress tensor (MST) an also compare the results to variations in the potential energy. As state in [17, p.66], the MST in the presence of a ielectric material is given by T = E D 1 (E D)I, (5) 2 where I is the rank-2 ientity. E is the total electrical fiel an D = ǫe the electrical isplacement, with ǫ = ǫ 0 ǫ r in the linear ielectric chuck material an ǫ = ǫ 0 in vacuum. We assume that the resultant force on any volume R if given by f = Tn. (6) R Across a surface of iscontinuity in T, this will give us the traction acting on the surface as [[T]]n. (7) Here we enote the jump [[T]] = T + T, where + inicate the values slightly above the surface in the n irection, or below the surface respectively. Assume now a geometry as shown in Fig. 1. By employing Gauss law, one can easily verify that the magnitue of the total electrical fiel is given by E = E(x)ˆx, with 0, x < 0 σ f E(x) = ǫ 0 ǫ r, 0 < x < σ f (8) ǫ 0, < x < +δ a 0, +δ a < x, where σ f is the free surface charge ensity on the conucting plates. The important MST component 0, x < 0 σf 2 T xx (x) = 2ǫ 0 ǫ r, 0 < x < σ (9) f 2 2ǫ 0, < x < +δ a 0, +δ a < x. Since in e-chucking applications we control the potential ifference rather than the surface charges, we use V = +δ a 0 Ex to erive σ f = ǫ 0ǫ r V +ǫ r δ a. (10) 3

5 For the case that δ a > 0, we then get by (7),(9), an (10) the electrostatic pressure acting on the mask back-sie as p = T xx (+δ a ) T xx (+δ a ) + = ǫ 0ǫ 2 rv 2 2(+ǫ r δ a ) 2, (11) which is equivalent to (1). For the pressure acting on the top-surface of the chuck ielectric which is equivalent to (2) an p = T xx () + T xx () = ǫ 0(ǫ 2 r ǫ r )V 2 2(+ǫ r δ a ) 2, (12) p c = T xx (0) + T xx (0) = ǫ 0ǫ r V 2 2(+ǫ r δ a ) 2 (13) for the pressure acting on the chuck electroe as state in (3). In a similar fashion we erive for the case δ a = 0 as picture in Fig. 2 the pressure p c,0 = T xx (0) + T xx (0) = ǫ 0ǫ r V (14) as given by (4). Note that (1)-(4) can also be erive by consiering the potential energy store in the total system capacitance which is relate to the electric fiel an the electric isplacement. As shown for example in [18, p.192], the energy of the system 1 can be obtaine by W = 1 D E. (15) 2 From this we fin using (8) an (10) the energy per unit area as w(,δ a ) = ǫ 0ǫ r V 2 2(+ǫ r δ a ). (16) For the case where L = +δ a is fixe, we efine the energy per unit area as ŵ() = w(,l ). One can then verify that which is equivalent to what we obtaine before. p = w(,δ a ) δ a (17) p = ŵ () (18) p c = w(,δ a) (19) p c,0 = w(,0), (20) 1 Here we refer to the energy associate with the separation of the free an boun charges, as well as the polarization of the molecules in the ielectric material.

6 5 F Mask p(δ a ) p (δ a ) p (0) F p c,0 = p(0) p (0) Y Y p(0) Y (a) (b) (c) Figure 3: Schematic of elastic chuck ielectric with stiffness Y, rigi mask an pressures acting uring(a) no-contact, (b) touch-own an(c) full-contact state. 3. Elastic Chuck Dielectric an Finite Element Bounary Conitions We now wish to iscuss the case where the mask an the chuck are in contact. As mentione earlier, two formulas for the pressure acting on the boiescanbefounintheliterature, namelyp c,0 = ǫ 0ǫ rv 2 anp(0) = ǫ 0ǫ r V In orer to explain the iffering factor of ǫ r, we consier the chuck ielectric as an elastic boy. Assume an elastic ielectric layer of the chuck with Young s moulus Y. We picture three ifferent states in Fig. 3. In Fig. 3(a) there is no contactbetweenthemaskanthechuckielectric, thevacuumgapisδ a, an we consier a stretch of the ielectric resulting from the pressure p (δ a ). 2 Assume that the mask approaches the chuck an is hel by force F to ensure static equilibrium. Just before the contact is establishe, the stretch in the chuck ielectric is measure as, which is picture in Fig. 3(b) an referre to as the touch-own state. When F = 0, the chuck ielectric is compresse by the pressure p c,0 = p(0) p (0) as shown in Fig.3(c). If we want to know the pressure that is necessary in orer to release the mask from the ielectric (i.e. bring the system back to the touch-own state), we require a force p(0) acting on the mask. The pressure p(δ a ) is also thetotalpressureactingonthechuckanthemaskwhenthereisnocontact. 2 In Appenix A we will argue that typical stretches of the ielectric are small an that it is not necessary to account for the variation of when calculating the electrostatic pressures.

7 6 Mask Mask Mask p(δ a ) p(δ a ) p(0) δ a p (δ a ) p(δ a ) p(0) p c (δ a ) Chuck Electroe Chuck Chuck (a) (b) (c) Figure 4: Loaing bounary conitions for finite element simulations: (a) gap-epenant pressure on mask back-sie an chuck ielectric/-electroe, (b) gap-epenant pressure on mask back-sie an chuck bounary surface, (c) constant pressure approximation. Thus it seems reasonable to use a simplifie finite element moel as picture in Fig. 4(b). Here we apply a pressure p epening on the vacuum gap δ a on the bounary surface of the chuck an mask back-sie surface. This allows us to avoi the moeling of a ielectric layer with the corresponing pressures as picture in Fig. 4(a). Furthermore, note that we can avoi a gap-epenant bounary conition formulation whenever δ a / 1 as we remark in Appenix A. As picture in Fig. 4(c), we then apply only the constant pressure p(0) on both surfaces. 4. Conclusions We calculate the electrostatic pressures for a Coulomb-type electrostatic chuck uner iealize conitions. In particular, we consiere the pressures acting on the mask back-sie, the chuck ielectric an-electroe an are able to explain the ifferences in the literature concerning the relation between the electrostatic pressure an the applie voltage. Finally, the implications for correct bounary conitions as use for example in finite element simulations are note. 5. Acknowlegements This work was inspire by Dr. Chytra Pawashe an Dr. Daniel Pantuso from the Intel Corporation, who pointe us to the ifferences in the literature. The authors wish to thank the Intel Corporation an Dr. Manish Chanhok for their generous an continue support of this research.

8 7 References [1] M. R. Sogar, a. R. Mikkelson, M. Nataraju, K. T. Turner, an R. L. Engelsta, Analysis of Coulomb an Johnsen-Rahbek electrostatic chuck performance for extreme ultraviolet lithography, Journal of Vacuum Science & Technology B: Microelectronics an Nanometer Structures, vol. 25, no. 6, p. 2155, [2] M. Nataraju, Analysis of the mechanical response of extreme ultraviolet lithography masks uring electrostatic chucking. PhD thesis, University of Wisconsin - Maison, [3] M. Nataraju, J. Sohn, S. Veeraraghavan, A. R. Mikkelson, K. T. Turner, an R. L. Engelsta, Electrostatic chucking for extreme ultraviolet lithography: Simulations an experiments, Journal of Vacuum Science an Technology B, vol. 24, pp , [4] S. Qin an A. McTeer, Wafer epenence of JohnsenRahbek type electrostatic chuck for semiconuctor processes, Journal of Applie Physics, vol. 102, no. 6, p , [5] E. Y. Shu, Optimization of electrostatic chuck for mask blank flatness control in extreme ultraviolet lithography, Proceeings of SPIE, vol. 6607, no. May 2011, pp J 66070J 12, [6] S. Veeraraghavan, J. Sohn, an K. T. Turner, Techniques to measure force uniformity of electrostatic chucks for EUV mask clamping, Proceeings of SPIE, vol. 6730, no. May 2011, pp G 67305G 11, [7] M. Nakasuji an H. Shimizu, Low voltage an high spee operating electrostatic wafer chuck, Journal of Vacuum Science & Technology A, vol. 10, no. March 1991, pp , [8] J. van Elp, P. Giesen, an A. e Groof, Low-thermal expansion electrostatic chuck materials an clamp mechanisms in vacuum an air, Microelectronic Engineering, vol , pp , June [9] A. G. Warly, Electrostatic Wafer Chuck for Electron Beam Microfabrication, Review of Scientific Instruments, vol. 44, pp , [10] J. P. Scott, Recent progress on the electron image projector, Journal of Vacuum Science & Technology, vol. 15, [11] R. War, Developments in electron image projection, Journal of Vacuum Science & Technology, [12] T. Watanabe, T. Kitabayashi, an C. Nakayama, Electrostatic Force an Absorption Current of Alumina Electrostatic Chuck, Japan Journal of Applie Physics, vol. 31, pp , [13] K. Yatsuzuka, F. Hatakeyama, K. Asano, an S. Aonuma, Funamental characteristics of electrostatic wafer chuck with insulating sealant, IEEE Transactions on Inustry Applications, vol. 36, no. 2, pp , [14] G. Kalkowski, S. Risse, G. Harnisch, an V. Guyenot, Electrostatic chucks for lithography applications, Microelectronic Engineering, vol , pp , Sept [15] J. Yoo, J.-S. Choi, S.-J. Hong, T.-H. Kim, an S. J. Lee, Finite Element Analysis of the Attractive Force on a Coulomb Type Electrostatic Chuck, Proceeings of International Conference on Electrical Machines an Systems, pp , [16] C. Pawashe, S. Floy, an M. Sitti, Multiple magnetic microrobot control using electrostatic anchoring, Applie Physics Letters, vol. 94, no. 16, p , [17] L. D. Lanau an E. M. Lifshitz, Electroynamics of Continuous Meia. Burlington, MA, USA: Butterworth-Heinemann, 2 e., [18] D. Griffiths, Introuction to Electroynamics. Upper Sale River, N.J.: Prentice- Hall, Inc., [19] G. Branstetter an S. Govinjee, Estimation of Particle Contamination Effects in Electrostatic Chucking, in submission, 2011.

9 Appenix A. Approximations We briefly justify the iealization of a rigi ielectric material for the electrostatic pressure calculation an iscuss the valiity of a constant pressure approximation as a further simplification as often use for example in finite element simulations. In orer to estimate the sensitivity of the electrostatic pressure versus the ielectric thickness, we linearize (1) at δ a = 0. By varying, we obtain the estimate p p = 2. (21) Consier a numerical example, where the mask an chuck are just in contact; i.e. δ a = 0. If we assume a typical ielectric constant ǫ r = 4, an an electrostatic pressure p = 15 kpa, we obtain by (2) a tensile force on the ielectric p = 11 kpa. Let us assume the Young s moulus of the material to be Y = 90 GPa. By taking the ratio, we estimate the relative change in the ielectric thickness as / = p /Y = 10 7, an then by (21), we obtain a relative change in the pressure of p/p = This is negligibly small an thus justifies the assumption of a constant ielectric thickness for the given example. Finally note the sensitivity of the electrostatic pressure calculation via (1) on the air gap δ a. By varying the air gap δ a, we obtain p p = 2ǫ r δ a 8. (22) In e-chucking applications, the peak-to-valley in the non-flatness of the mask back-sie correspons to δ a in the iealize moel. From (22) we see that whenever δ a / 1, the resulting pressure variation is negligible. Thus, as a further simplification of the moel as propose in Section 3, it is reasonable to assume a constant pressure of magnitue p = ǫ 0ǫ 2 rv (23) acting on the mask back-sie an the chuck, whenever δ a / 1. This was also observe in [19], where small ratios δ a / woul not alter the preiction of the mask eformation when comparing a gap epenant pressure formulation via (1) an the constant pressure approximation (23).