IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 42, NO. 3, MARCH 2014 729 Analytical Determination of Collisional Sheath Properties for Triple Frequency Capacitively Coupled Plasma M. T. Rahman and M. N. A. Dewan Abstract—A self-consistent analytical model for a time- independent collisional capacitively coupled plasma (CCP) sheath driven by a triple frequency (TF) RF current source is proposed. Sheath parameters are calculated using this model for some standard plasma parameters and are compared with those of a single frequency (SF) and a dual frequency (DF) capacitively coupled collisional sheath. This model estimates higher values of sheath width and potential with more oscillating behavior compared with SF and DF sheaths. By proper choice of source frequencies or phase differences in the source currents, it is possible to adjust the ion energy hitting the electrode. Use of TF source is found to facilitate better control upon sheath parameters for collisional CCP. Index Terms—Capacitively coupled plasma (CCP), collisional sheath, RF sheath, triple frequency (TF). I. I NTRODUCTION C APACITIVELY coupled plasma (CCP) has been widely used in plasma-assisted processing, such as plasma etching, deposition, and other surface treatments in micro- electronics industries. Multi-frequency driven CCP holds a great promise for next generation etching devices and has received an intense investigations in recent years [1]–[23]. Many models, theoretical studies, and experimental analyses have investigated the extent to which the plasma density and ion energy are controlled independently by a high-frequency (hf) and a low-frequency (lf) source, respectively [1]–[23]. Such flexibility is highly desirable for certain processes in semiconductor industries. Robiche et al. [1], Jiang et al. [2], Guan et al. [3], and Dai et al. [4] made some theoretical and simulation based work on collisionless capacitive plasma energized by dual frequency (DF) source considering different sets of assumptions. Kim et al. [5] and Lee et al. [6] introduced the concept of an effective frequency for DF CCP and triple frequency (TF) CCP, respectively, based on the assumption of homogeneous ion distribution within the sheath. Along with the collisionless sheath, study of collisional sheath for multi-frequency cases is also important to under- stand the physics of capacitive discharge in intermediate pressure range. Boyle et al. [7] developed an analytical model to investigate the nonlinearity in an inhomogeneous collisional Manuscript received July 8, 2013; revised January 9, 2014 and January 26, 2014; accepted January 31, 2014. Date of publication February 19, 2014; date of current version March 6, 2014. The authors are with the Department of Electrical and Electronic Engi- neering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh (e-mail: rumman_tanjidur@yahoo.com; dewan@eee.buet.ac.bd). Digital Object Identifier 10.1109/TPS.2014.2304463 sheath for DF CCP, focusing on the sheath voltage and width. Boyle et al. [8], [9] numerically studied the possibility of controlling the ion energy and ion flux on the electrode for collisional DF sheath. To describe the sheath dynamics over a wide frequency range, several semianalytical models were also developed for DF sheath considering the time-dependent equations. Rahman et al. [10] developed a time-dependent self- consistent analytical model for DF collisional sheath. In their model, the time-dependent terms in ion fluid equations are ignored. Recently, it is reported that the quality of separate control of ion energy and flux in DF discharge is limited due to the coupling of two frequencies and effect of secondary electrons [24]–[26]. Introduction of an intermediate frequency may solve the problem. Using three RF frequencies, it should be possible to achieve the desired independent control over ion flux on the electrode and the ion bombarding energy. It is also evident in [6], [11], and [27]. None of these works considered the intermediate or high pressure ranges where ion-neutral collisions dominate and the property of plasma is quite different from that of collisionless case. Collisionless sheath model is valid only for low pressure ranges, around a few millitorrs, where the ion mean free path is larger than the sheath width. Therefore, a proper collisional model is necessary to determine the sheath properties in TF CPP. This paper proposed an analytical model for TF time-independent collisional capacitive sheath considering inhomogeneous ion distribution. The sheath parameters are then calculated using this model and compared with those of a single frequency (SF) collisional model proposed in [28] and a DF collisional model proposed in [7]. The effects of source parameters—currents and frequencies—are also investigated. II. ANALYTICAL MODEL In this model, we considered current source having three RF frequencies—lf, mid-frequency (mf), and hf, applied to the electrode of a CCP chamber. Due to RF source, an oscillating RF sheath will be formed near the electrode. Fig. 1 shows a schematic diagram of the RF sheath. Here, we assumed a collisional sheath where ion motion is one dimensional. Ions crossing the ion sheath boundary at x = 0 with Bohm velocity, u B = (eT e / m i ) 1/2 , are accelerated within the sheath and strike the electrode at x = s m , where s m is the maximum sheath thickness. Here, s (t ) is the instantaneous electron sheath boundary. The bulk plasma is assumed to have a density n 0 0093-3813 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.