Surface-wave propagation along a corrugated plasma–dielectric interface

Surface & Coatings Technology 200 (2005) 792 – 795 www.elsevier.com/locate/surfcoat

Surface-wave propagation along a corrugated plasma–dielectric interface Ivan P. Ganacheva,b,T, Hideo Sugaib a

Shibaura Mechatronics Corporation, 2-5-1 Kasama, Sakae-ku, Yokohama 247-8560, Japan Department of Electrical Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

b

Available online 24 March 2005

Abstract Recently it has been found that the performance of surface-wave (SW) sustained microwave discharges can be enhanced by corrugating the dielectric–plasma interface guiding the SW. The present contribution explains this behavior in terms of the SW dispersion as modified by the presence of corrugations. The latter are treated as an anisotropic dielectric medium with effective permittivity tensor governed by the corrugation depth, pitch and dielectric loading ratio 0 b c b 1. Closed form SW dispersion equations are introduced. We demonstrate that the corrugations significantly lower the threshold electron density above which SW propagation is possible. Moreover they can reduce the SW group velocity and thus improve the wave–plasma interaction. Additional advantage is that one can custom-tune the SW dispersion and the wave–plasma interaction by controlling the dielectric loading ratio c. D 2005 Elsevier B.V. All rights reserved. PACS: 52.80.Pi (High-frequency and RF discharges); 52.35.Hr (Physics of plasmas and electric discharges—electromagnetic waves); 52.65.Kj (Plasma simulation—magnetohydrodynamic and fluid equation) Keywords: Surface-wave plasma; Surface-wave dispersion; Corrugated surface; Plasma processing; FDTD electromagnetic simulation; Backward waves

1. Introduction SW discharges established themselves as reliable, clean and flexible plasma sources for semiconductor and other surface processing. The plasma in a SWP (surface-wave plasma) source is sustained by an electromagnetic surface wave propagating along the interface between an overdense non-magnetized plasma and some dielectric body. The field of the surface wave is negligible deep in the plasma (at distances longer than the skin-depth d, which is of the order of 0.5–3 cm in most processing plasmas). The two major configurations are the cylindrical SWP column [1,2] and the planar SW discharge [3], with various other configurations finding also their specific applications [4,5]. In a typical SWP column the SW is excited by a launcher (usually a coaxial surfatron [1] or rectangular-waveguide surfaguide [2]) and then it gradually transfers its power to the plasma sustained in a dielectric tube, until the power is totally

absorbed at some distance from the launcher, where the plasma ends. The planar geometry is suitable for large-area processing (e.g. semiconductor wafers). Here one wall of the discharge chamber is made of a large dielectric plate and the SWP is sustained below it, while the launcher is on top of the plate. Various launchers (mostly slots) have been applied in two major concepts: traveling wave discharges with distributed excitation [3–6] and locally launched standing-wave discharge [7,8]. Fig. 1 shows the SW phase curve (dependence of SW wave number b on the electron density n e expressed via the normalized electron plasma oscillation period g = x / x pe at fixed wave frequency x) for the simplest case of a noncorrugated planar interface between a semi-infinite perfect dielectric with permittivity e d and semi-infinite homogeneous plasma of density n e and electron plasma frequency x pe. SW propagation is possible only if the electron density is higher than the threshold value of the bSW resonance densityQ nSW unc ð1 þ ed Þ;

T Corresponding author. Shibaura Mechatronics Corporation, 2-5-1 Kasama, Sakae-ku, Yokohama 247-8560, Japan. E-mail address: [email protected] (I.P. Ganachev). 0257-8972/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2005.02.090

ð1Þ

where n c is the cut-off plasma density. The existence of this threshold is an unwelcome limitation for processing plasmas:

I.P. Ganachev, H. Sugai / Surface & Coatings Technology 200 (2005) 792–795

Normalized wavelength λ/λ0 1.0 0.5

Normalized electron plasma oscillation period ω/ωpe

0.1 5

SW resonance

0.4

(1 + εd)−1/2

0.3

εd = 4 10 20

0.2

0

ε d = 10

εd

0.1

0

100

2 4 6 8 Normalized wave number β/k0

Normalized electron density ne/nc

0.5

0.2

10

Fig. 1. Typical SW phase diagrams along a non-corrugated plasma– dielectric interface.

for the typical case of x / 2k = 2.45 GHz and e d c 4 (quartz) it requires electron densities of at least 3.7  1011 cm 3. Many processes do not require and even forbid such high electron densities. Another problem for the operation of SW discharges is the rather weak wave–plasma interaction at high electron densities. The SW has to travel over a long distance in order to transfer significant portion of its power to the plasma. This complain can be found, e.g., in section 13.3 of the classical textbook by Liebermann and Lichtenberg [9]. The SW absorption length a  1 (defined as the inverse of the SW attenuation coefficient a) is the measure for bhow efficientQ the wave–plasma interaction is. It gives the distance over which the SW transfers 1  e 2 c 86.5% of its power to the plasma, or, in other terms, the distance at which the SW amplitude decreases e = 2.718. . . times. The SW absorption length is directly related to the slope of the SW phase curve via the relation [10] a1 ¼

2 d 2 xg c mg ; m d m

793

by the plasma (see, e.g. Ref. [11], where a 12 m long SW discharge is reported); (b) operating slightly above the SW resonance density (1) and tolerating the electron density restrictions imposed; (c) controlling the position of the SW resonance n c (1 + e d) by choosing materials with various permittivities [12]); (d) choosing a wave frequency different from 2.45 GHz, which changes n c and, thus, n SW [13]; (e) applying reflections at the plasma edge, thus permitting the SW to traverse the plasma many times effectively increasing the interaction length. The last method can be applied only at a discrete set of electron densities that tune the finite plasma with the reflections at the edge to some of the various possible standing wave modes. This has the disadvantage that continuous plasma density control becomes impossible: increasing the applied microwave power results in a series of a density jumps between the various standing-wave modes [14,15]. Recently Yamauchi et al. demonstrated that at least two of the aforementioned problems (the density jumps and the minimal threshold electron density for SW propagation) in a 2.45 GHz SW plasma can be overcome by corrugating the dielectric–plasma interface guiding the SW [16]. Preliminary results suggest an optimum value of about 0.5 for the dielectric loading ratio of the corrugation (the ratio c = s / q of the dielectric ridge width s to the corrugation pitch q, see Fig. 2). However, up to now there has been no clear understanding of the role played by the corrugations, and it has not been clear why corrugations are beneficial in some cases and not in others. In particular, there has been no explanation as to why a dielectric loading ratio c c 0.5 is optimal. This contribution attempts to explain this behavior by analyzing the modification of the SW dispersion introduced by the corrugations.

2. Effective permittivity tensor ð2Þ

where m is the electron-neutral collision frequency for momentum transfer. Indeed Eq. (2) gives the distance covered by a SW pulse traveling at the group velocity m g for the SW decay time 2 / m while the amplitude decays e = 2.718. . . times. From Eq. (2) it is obvious that there is little we can do to influence the intensity of wave–plasma interaction in plasma processing applications: The wave frequency x is usually fixed by the equipment. The gas pressure and composition, and thus the collision frequency m, are imposed by the process requirements. The only free control parameter is the position g along the phase curve, which controls the group velocity. However, using this to control the SW absorption is at the cost of changing the electron density, too: independent control of electron density and SW absorption is not possible. To overcome this problem there have been several approaches: (a) making the SW discharge very long to provide for enough space for the SW power to be absorbed

If the SW length k is much longer than the corrugation pitch q, one can treat the corrugations as a homogeneous dielectric with an effective dielectric constant between the

s

ρ

dielectric

d

Longitudinal propagation

Transverse propagation

plasma

Fig. 2. SW propagation along a corrugated plasma–dielectric interface.

794

I.P. Ganachev, H. Sugai / Surface & Coatings Technology 200 (2005) 792–795

1

permittivities of the plasma e p and the dielectric e d. The actual value should depend on the loading ratio c: the value e eff = c e d + (c  1) e p was proposed in Ref. [17]. A more rigorous analysis shows that the effective permittivity of the corrugations is not a scalar, but a tensor: in direction normal to the corrugated interface the effective permittivity is indeed as proposed in Ref. [17] et ¼ ced þ ð1  cÞep :

0.8

ω ωpe

0.6

0.4

ð3Þ

However, for field components parallel to the plasma– dielectric interface the effective permittivity differs along and across the corrugations: along the corrugations it is still as given by Eq. (3), while across the corrugations the expression is 1=e8 ¼ c=ed þ ð1  cÞ=ep :

0.2

0

We consider a surface mode propagating with a wave number b along the corrugated interface between a semiinfinite plasma and a semi-infinite dielectric of permittivities e p and e d, respectively. The interface is in the xz plane, the wave propagates along the z axis, and the corrugation depth is d (Fig. 2). For a TM surface mode the dispersion equation is

4. Phase curves An example for transverse SW propagation across the corrugations (in accordance with the excitation in the experiments [16]) is shown in Fig. 3a. The classical SW phase diagram for non-corrugated surface (the dotted lines in Fig. 3) has a single branch extending from x / x p = 0 up to x / x p = (1 + e d) 1/2, i.e. electron density from n e = l down to the surface-wave resonance density n SW = n c (e d + 1). Corrugations modify dramatically the phase diagram: instead of a single curve we get many branches in four clearly distinguished areas: a stop band and a single curve for



cp =ep þ cd =ed þ cp cd =ep ed þ c2c =e2cz ð5Þ

where c p,d = (b 2  e p,d k 02)1/2 are the transverse (away from the interface) SW decayconstants in the plasma and the dielectric, respectively, c c = (b 2 e cz / qcy  e cy k 02)1/2 is the transverse propagation constant in the corrugation layer, k 0 = x / c is the free-space wave number, e cy and e cz are the components

1

(b) longitudinal

0.8

N = 1 + ε d γ /(2 − γ )

0.6

N = 1 + εd

N = 1 + ε d N = 1 + ε d (1– γ ) / γ N = 1 + ε d γ /(1 − γ )

2

5

N = 1 + ε d (1 + γ ) /(1 − γ )

10

0.2 100 0

2

6

10

14

18 2

6

10

14

18

Normalized electron density N = ne/nc

(a) transverse

0.4

6

of the effective permittivity tensor in the corrugation along the y and z axes, respectively. For propagation parallel the corrugation ridges (denoted as longitudinal in Fig. 2) e cy = e cz = e ||, where e || is given by Eq. (3). For transverse propagation e cy = e ||, but e cz = e 8 , where e 8 is given by Eq. (4).

3. Dispersion equation

ω ωpe

5

4

Fig. 4. Comparison of analytical (solid curves) and numerical FDTD results (squares) for e d = 4, c = 1 / 2, x / 2k = 2.45 GHz, d = 5 mm. The dotted curve represents propagation along a non-corrugated interface.

The last two expressions are derived by regarding the corrugations as many dielectric- and plasma-filled capacitors connected in parallel [for Eq. (3)] and in series [for Eq. (4)].

1

3

Normalized wave number β/k0

ð4Þ

ðecz =cc Þtanhðcc d Þ ¼ 0;

2

Normalized wave number β/k0 Fig. 3. SW phase curves for transverse (a) and longitudinal (b) propagation for e d = 4, c = 2 / 3, x / 2k = 2.45 GHz, d = 5 mm (solid curves) compared to propagation along a non-corrugated interface (dotted curves).

I.P. Ganachev, H. Sugai / Surface & Coatings Technology 200 (2005) 792–795

intermediate electron densities, where c c2 N 0, and two multicurve c c2 b 0 zones below and above it. The phase curves in the lower high-density multi-curve region feature negative slope corresponding to backward wave propagation. The case of longitudinal SW propagation in direction along the corrugation ridges is shown in Fig. 3b: one gets two phase curves similar to the classical curve for noncorrugated interface. However, the resonance densities are modified from the classical n c (e d + 1) to two new values: one higher and one lower than n c (e d + 1) [17]. Again, one can control the actual positions of those resonances by adjusting the dielectric loading ratio c. To validate the effective-permittivity approach Eqs. (3) and (4), on which the dispersion Eq. (5) is based, we compared the phase curves to results of 2D and 3D FDTD [18] numerical simulations which make use of neither Eqs. (3) nor (4) (the squares in Fig. 4). The agreement is very good, except for short waves, where the requirement k J q is not met.

5. Discussion and conclusion The problems of threshold plasma density and too weak wave–plasma interaction are at least partially solved by applying corrugations, as visible from the phase diagrams in Fig. 3. First, the corrugations permit SW propagation down to much lower electron densities (up to e d + 1 times lower than the non-corrugated case). Second, the abundance of phase curves greatly relieves the problem of large standingwave mode jumps: changing the electron density results in series of small non-disruptive almost continuous transitions to the next available mode. Additional relief is provided by the increased SW absorption for high densities as indicated by the lower slope of the phase curves there: it enables the generation of homogeneous standing-mode-free SWP at higher densities than without corrugations. And finally, all these features can be custom-tuned by varying the dielectric

795

loading ratio c. In particular, for SW propagating at 908 to corrugations (transverse propagation as indicated in Fig. 2) with c = 0.5 the stop band above the middle-region single curve (and that curve itself) disappears, which may be why c close to 0.5 is optimal.

References [1] M. Moisan, C. Baudry, P. Leprince, IEEE Trans. Plasma Sci. PS-3 (1975) 55. [2] M. Moisan, C. Baudry, L. Bertrand, E. Bloyet, J.M. Gagne´, P. Leprince, J. Marec, G. Mitchel, A. Ricard, Z. Zakrzewski, Gas Discharges (IEE Conference Publication) 143 (1976) 382. [3] K. Komachi, S. Kobayashi, J. Microw. Power Electromagn. Energy 24 (1989) 140. [4] D. Korzec, F. Werner, R. Winter, J. Engemann, Plasma Sources Sci. Technol. 5 (1996) 216. [5] E. Bluem, S. Be´chu, C. Boisse-Laporte, P. Leprince, J. Marec, J. Phys. D: Appl. Phys. 28 (1995) 1529. [6] M. Kanoh, K. Aoki, T. Yamauchi, Y. Kataoka, Jpn. J. Appl. Phys. 39 (2000) 5292. [7] M. Nagatsu, G. Xu, I. Ghanashev, M. Kanoh, H. Sugai, in: T. Makabe (Ed.), Proc. 13th Symposium on Plasma Processing, Tokyo, January 29–31, Japan Society of Applied Physics, Tokyo, 1996, p. 9. [8] I.P. Ganachev, H. Sugai, Plasma Sources Sci. Technol. 11 (2002) A178. [9] M.A. Liebermann, A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, Wiley, New York, 1994. [10] I. Ghanashev, A. Shivarova, H. Schlqter, Radio Sci. 29 (1994) 1265. [11] S. Nonaka, Jpn. J. Appl. Phys. 33 (1994) 4226. [12] M. Furukawa, N. Saito, K. Kawamura, T. Koromogawa, H. Shindo, Jpn. J. Appl. Phys. 37 (1998) L1005. [13] M. Nagatsu, A. Ito, N. Toyoda, H. Sugai, Jpn. J. Appl. Phys. 38 (1999) L679. [14] I. Ghanashev, M. Nagatsu, G. Xu, H. Sugai, Jpn. J. Appl. Phys. 36 (1997) 4704. [15] I. Ghanashev, H. Sugai, Phys. Plasmas 7 (2000) 3051. [16] T. Yamauchi, E. Abdel-Fattah, H. Sugai, Jpn. J. Appl. Phys. 40 (2001) L1176. [17] I. Ganachev, H. Sugai, Surf. Coat. Technol. 174–175 (2003) 15. [18] K.S. Yee, IEEE Trans. Antennas Propag. AP-14 (1966) 302.