Measurement and Error Evaluation of Electrical Parameters at Plasma Relevant Frequencies and Impedances y Craig Garvin, Brian E. Gilchrist, Dennis S. Grimard, and Jessy W. Grizzle var Abstract var : measured value of var var The errors present in electrical measurements at frequencies and d : estimated value of var impedances relevant to plasma processing in the semiconductor var var = var : relative error in as var industry are studied. A theoretical bound on calculated deliv- var var = var var var ered power error as a function of measured electrical values is var : absolute error in as derived. The derivation shows that for constant measurement er- var j var ror, power error is a linear function of load impedance expressed j : absolute value of (rms if time varying) in terms of voltage standing wave ratio. This bound is supported ? var var : complex conjugate of by experimental data taken with both a directional coupler and a var var voltage- and current- based probe. Linear and nonlinear model < : real part of based sensing methods are implemented which reduce power var var = : imaginary part of error by a factor of five over a standard calibration. Published results are cited which indicate that the voltage standing wave P ef r : power at ref point in circuit ratio of typical plasma processing and experimental regimes is + V : complex value of forward voltage at ref point in circuit ef high enough to cause small measurement errors to result in large r calculated power errors. V : complex value of reverse voltage at ref point in circuit ref ef Nomenclature r : complex reflection coefficient at ref point in circuit : phase angle of reflection coefficient at ref point in circuit ef electrical state: electrical state, no representation chosen r v ef r : complex value of voltage at ref point in circuit load: a one port network with a specific impedance or reflection i ef coefficient, no representation chosen r : complex value of current at ref point in circuit Z ef probe: measured values taken at the probe’s measurement ports r : complex impedance at ref point in circuit Z : phase angle of impedance at ref point in circuit ef srp: ‘sensor reference plane’ the reference plane inside the sen- r sor where measurement actually takes place. See Sec- S dut : scattering matrix of device under test dut tion 6.1 s c r : scattering parameter r,c of scattering marix lrp: ‘load reference plane’- the reference plane at the point in A the circuit where a load is connected to the probe. See Fig- dut : ABCD matrix of device under test dut ure 6 x ref :electrical state at ref point in circuit, no representation 1 Introduction chosen In semiconductor manufacturing applications, RF powered + V (V , ): electrical state represented as forward and reverse :56MHz plasma processes are typically driven at 13 . It has been voltage waves well known since the early days of plasma processing that the power delivered to the plasma is a dominant factor affecting pro- i ( v , ): electrical state represented as current and voltage cessing. More recent published data [1–4] indicates that any- var var 90 ’( )- based’ : electrical state measured as ‘ ’ where between 10 and of the input power is dissipated outside the plasma discharge. Such a finding motivates using var var : actual value of the delivered power and not the generator power as the control C. Garvin, D. Grimard, and J. Grizzle are with the University of Michi- input in plasma processing, as done by [5, 6]. Clearly, control- gan Electronics Manufacturing Laboratory, 3300 Plymouth Rd., Ann Arbor, MI ling the delivered power requires determining this value with 48105-2108 y Brian E. Gilchrist is with the University of Michigan Radiation Laboratory, small enough errors that using measured delivered power as a 1301 Beal Ave. Ann Arbor, MI 48109-2122 feedback variable actually improves performance. 1 2 Garvin et al Measurement and Error Evaluation of Electrical Parameters Another goal of measuring plasma electrical parameters is state refers the specification of either complex current and volt- i to reconstruct physical plasma quantities from these electrical age, ( v , ), or complex forward and reverse voltage waves, + V measurements. Many researchers [1, 7–13] have proposed mod- (V , ). A reference plane is a plane perpendicular to the els relating plasma physics to observed impedance. Although direction of power flow in a circuit at which electrical state can varying in the specifics, most agree that the plasma sheath can be specified. One of the main reference planes we will use is be represented by a capacitor, and the rate of dissociation of the load reference plane. This reference plane refers to the point electrons from atoms as a resistor. If we seek to determine the in the circuit where a load is connected to the probe system, as plasma parameters from measured electrical parameters, then seen in Figure 6. In all equations, if no reference plane is spec- error free electrical measurements are also desired. ified, then the equation is true as long as all variables are at the It has long been known that the plasma acts as a nonlinear same reference plane. Both voltage, current and forward, re- load, to some degree rectifying the input frequency and result- verse voltage representation of electrical state are used, related ing in a DC offset and harmonics of the fundamental frequency. by (1). Recently, Klick [14] has proposed a model for interpreting these + 1 1 V higher harmonics as additional indicators of the plasma’s physi- v = 1 1 (1) i V Z Z cal state. It is likely that whatever challenge exists in measuring :56MHz the plasma electrical state at 13 is exacerbated at higher The following derived quantities are obtained for power being harmonics of this frequency. This follows from the observation delivered at a particular reference plane in a circuit. that the major sources of error: losses, radiation, and reference 2 2 plane changes all become increasingly prominent as frequency + j V j jV j = increases. P (2) Z Numerous measurements of plasma electrical parameters ? = <v i P (3) have been published [3, 12, 15–20]. All of these note the mea- surement equipment used and describe the calibration methodol- Note that all electrical quantities are specified as rms values, ogy implemented. Clearly, small errors in measuring electrical so no factor of 1 is required in power calculation 2 + V v i parameters such as (V , )or( , ) are essential to deter- minating delivered power and plasma impedance with small er- j j rors. This paper addresses power and impedance measurement 1+ VSWR = (4) j j at plasma relevant frequencies and impedances. A theoretical 1 derivation of power error as a function of measurement error is v = Z (5) presented. Power error is shown to be a linear function of volt- i age standing wave ratio for constant measurement error. Exper- V = (6) imental data is presented to support the theoretical limit. + V P Relatively high errors in lrp (power at the load reference Z Z load = ad plane) are observed even under benign test conditions. Sec- lo (7) Z + Z ad tions 5.2 and 6.3 develop methods for reducing the error of RF lo probes by model based sensing. Tests of two high power probes dut input are presented representing a typical directional coupler and volt- dut output age and current based probe. To minimize errors, experiments + Reactive V 1 S22 are performed at a single frequency and very low power on sim- Load S11 ulated plasma loads. Extension to actual plasmas and plasma- S21 S relevant power levels is straight forward. Additional results cov- 12 + Port 1 Reference Plane V 2 2 ering real time and high power compatible data acquisition sys- 3 s s 11 12 6 tems will be covered in a subsequent paper. 7 4 5 Port 2 Reference Plane S = dut s s 21 22 2 Microwave and Network Theory Figure 1: Determining the scattering parameters of a two port Summary network Two port representations are used to identify network ele- This section summarizes the basic microwave and network the- ments and to transform between reference planes. The scatter- ory needed to read this paper, as the experimental work and re- ing matrix is used for direct identification of a network. Fig- sults presented in this paper rely upon these concepts. The mate- ure 1 shows a typical application. The device under test, dut,is rial is derived from [21] and can be obtained from any standard connected between the two ports of the vector network analyzer text in microwave engineering. As a simplification, all equations S which measures the scattering matrix: dut . The scattering ma- Z assume a constant characteristic impedance ( ) throughout the trix relates forward and reverse voltages according to (8). circuit. + V s s The electrical quantities in a circuit will be referred to as V 11 12 1 1 = + (8) s s V the ‘electrical state’ata‘reference plane’. The electrical V 21 22 2 2 accepted by J. Vac. Sci. Tech., Vol. 16, No. 2, Mar/Apr 1998 3 Transforming between reference planes is achieved with the We define measurement error as follows: The measurement + V j jV j ABCD matrix, as shown in Figure 2. The ABCD matrix is errors are fixed values: j and independent of the S uniquely determined by dut and allows the voltage and current magnitude of measured quantity. It is obvious that as the ratio of + + j V j V j at two reference planes to be related by (9). j to changes, so do errors in power. Accordingly, it is reasonable to derive a relationship between measurement + v A B v + j V j 1 2 V j errors and power errors for a fixed ratio of j to = (9) i C D i 1 2 defined in (12).