A simple model of solenoidal inductively coupled plasma sources considering finite size

J J Lee,S J Kim,Y S Lee,K K Kim,S J You

doi:10.1063/1.5133862

Abstract

This paper proposes an updated transformer model for solenoidal inductively coupled plasma sources that can be applied even in low electron density regions. The proposed model can handle plasma in a finite geometry where the electric fields propagating from each boundary overlap, employing a simple analytic expression of the electric field, a one-dimensional (1D) sine hyperbolic function. Based on this field expression, all circuit elements of the transformer model that depend on the electron density, namely, plasma resistance, magnetic inductance, and mutual inductance, can now be obtained. Comparison of absorbed power as well as the circuit elements calculated using the proposed model, named here as the 1D transformer model, shows good agreement with the electromagnetic model, which is known for being quite accurate for cold plasma not only in high but also in low electron density regions. Results also indicate that the 1D transformer model is not limited to a specific setup but rather can be applied in a wide range of discharge conditions.

Highlights

The advantages of inductively coupled plasma (ICP) are numerous, including high plasma density at low pressure, reduction of ion damage, and independently controllable ion energy.1–3 compared with other plasma sources, such as helicon and electron cyclotron resonance, ICP sources can be scaled up to accommodate larger wafer sizes.4 By virtue of these merits, ICP sources are widely used in various plasma processing industries
The impedance of the plasma source and the efficiency of power transfer to plasma can be estimated,11 while handling the ICP source as an electric transformer having a resistive property coupled with rf power
Model only in the high electron density region; on the other hand, regardless of the electron density, the 1D transformer model is in good agreement with the EM model despite its simple assumption of expressing the electric field as a hyperbolic sine function

Summary

INTRODUCTION

The advantages of inductively coupled plasma (ICP) are numerous, including high plasma density at low pressure, reduction of ion damage, and independently controllable ion energy. compared with other plasma sources, such as helicon and electron cyclotron resonance, ICP sources can be scaled up to accommodate larger wafer sizes. By virtue of these merits, ICP sources are widely used in various plasma processing industries. With the assumption that the penetration depth of the fields (skin depth δ) is much shorter than the chamber size (R), the effective geometry of a plasma current loop can be readily determined, and the circuit elements that depend on the plasma parameters can be calculated using the geometry parameters.10 From these circuit elements, the impedance of the plasma source and the efficiency of power transfer to plasma can be estimated, while handling the ICP source as an electric transformer having a resistive property coupled with rf power. The high electron density region; there is a discrepancy between the values in the low-density region, where the absorbed power of the EM model shows a strong dependency on the electron density, but that of the transformer model does not This result reflects that the previous transformer model is applicable to highdensity but not low-density plasma. The absorbed power is calculated and compared with that of the EM model under various conditions of pressure and operating frequency

MODEL DESCRIPTION

RESULTS AND DISCUSSION

CONCLUSION