Comments on plasma diagnostics with microwave probes Valery Godyak

Citation: Physics of Plasmas 24, 060702 (2017); doi: 10.1063/1.4984781 View online: http://dx.doi.org/10.1063/1.4984781 View Table of Contents: http://aip.scitation.org/toc/php/24/6 Published by the American Institute of Physics PHYSICS OF PLASMAS 24, 060702 (2017)

Comments on plasma diagnostics with microwave probes Valery Godyaka) University of Michigan and RF Plasma Consulting, Brookline, Massachusetts 02446, USA (Received 8 March 2017; accepted 8 May 2017; published online 5 June 2017) Analysis of recent publications on microwave probe diagnostics shows that some assumptions used in microwave probe models are unrealistic and ambiguous, which puts into question the validity of those diagnostics. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4984781]

A variety of microwave probes, MP or active plasma A more sophisticated approach to find e from the measure- resonance spectroscopy, APRS, have been recently proposed ment of microwave probe reactance was shown in Ref. 5. as new tools for plasma diagnostics.1–13 The microwave Except the hairpin probe, invented by Stenzel10 and later probe diagnostics are based on the resonance response in the refined in Ref. 13, all recently proposed microwave probes absorption or reflection spectrum of some electro-dynamic are reincarnations (with some modifications) of the half cen- structure (probe) immersed into a plasma. Depending on the tury old resonance probe proposed by Takayama et al.,16 see probe structure and particular resonance mode, the probe also Ref. 17, and of the plasma resonance (cut off) probe pro- 18 resonance frequency, xr, is some modeled function of the posed by Levitskii and Shashurin. But, in spite of many plasma frequency, xpe, corresponding to the local plasma theoretical and experimental efforts to refine these micro- density; xr ¼ xr(xpe,Te). wave probe methods, neither of them became a routine diag- Different MPs with different probe structures, measure- nostic tool. ment techniques, and models coupling probe resonance fre- In our opinion, the old microwave probe methods could quency with plasma frequency have been analyzed for not be used as diagnostic tools because of the many uncer- inferring plasma density, n, electron temperature, Te, and tainties and unrealistic assumptions in theories and models electron collision frequency, e. for inferring plasma parameters from measured microwave Examples of existing microwave probes are given in probe characteristics. We are not aware of any existing criti- recent papers.4,8 They can be divided into three categories: cal analysis of microwave probe validity. The purpose of this article is to bring to attention some problems related to (1) Cutoff probe based on the minimal microwave signal microwave probes found in the current literature, which puts transmission from one electrode (probe) to another at into question their viability as new diagnostic tools. x ¼ x . r pe Any kind of plasma probe diagnostics (Langmuir, B-dot, (2) Hairpin probe (and its modifications) based on the reso- and microwave probes) implies an inferring of plasma local nance frequency shift of the probe resonance frequency parameters not distorted by the probe presence. Insertion of a with (x ) and without (x ) a plasma; x 2 ¼ x 2 x 2. r 0 pe r 0 probe, however, can lead to disturbances in the plasma den- (3) Resonance probe or plasma absorption probe (and many sity, electron temperature, plasma current, and ionization bal- new names for the same) based on the resonance in some ance. The conditions for neglecting those distortions are well surface mode of the plasma-sheath-probe structure. In known for Langmuir14 and, partly, for B-dot19 probes. the simplest case of a spherical probe, for an electrostatic Generally, the resonance frequency of a microwave probe symmetrical mode, x 2 ¼ x 2(1 þ R/S), where R is the pe r depends on the plasma density, probe geometry, electron tem- probe radius and S is the thickness of the sheath around perature, and sheath thickness. For a cut-off probe in the first the probe. This resonance is the well-known series category mentioned above, the sheaths around the transmit- sheath-plasma resonance when the sheath capacitive ting and receipting probes and their holder reduce the plasma reactance is compensated by the plasma inductive reac- density between closely set probes. The probe sensitivity and tance at x < x . pe accuracy deteriorate when the distance between the probes One attractive feature of microwave probes is their becomes comparable to the sheath thickness. For a hairpin immunity to probe surface contamination that may prevent probe in the second category, the presence of the sheath using Langmuir probes (LPs) and electrostatic analyzers in around the hairpin resonator affects its resonance frequency plasma processing reactors. Note that the probe contamination since the dielectric constant of the sheath is larger than the problem in plasma processing reactors was successfully dielectric constant of the surrounding plasma.13 For a reso- resolved using contemporary Langmuir probe techniques.14,15 nance probe in the third category, the sheath affects directly It is believed that another attractive feature of micro- the resonance frequencies, and in contrast to previous cases, wave probes is their ability of inferring the electron collision cannot be considered as a relatively small secondary effect. frequency, e, by measuring the resonance probe Q-factor, When a probe and its holder are inserted into a plasma, Q ¼ xr/e, or the width of the resonance peak, Dx ¼ e. they inevitably cause the plasma density depletion around the probe and its holder. Plasma depletion around the probe a)[email protected] is similar to plasma depletion near the plasma chamber wall.

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Such plasma disturbance inflicted by a Langmuir probe has yields the Bohm criterion that the ions enter the sheath with been studied theoretically and in experiments for collisional the ion-sound speed vs. This model gives an infinite sheath plasmas.20,21 Those studies showed a considerable drop in (k ¼1) and is obviously not suitable for sheath width evalua- plasma density around the probe at the distance nearly equal tion.25 To find the sheath width some authors use the steady to a few probe radii. On the other hand, the microwave field state Childs-Langmuir law or other simplified sheath mod- induced by the probe is localized in the same plasma els.26 All those steady-state models give different sheath depleted zone on a scale comparable to the probe radius R. widths, which are qualitative estimates, and therefore differ- Therefore, the microwave probe senses a depleted plasma ent coefficients k. However, none of them is suitable for the density n* < n0 and the sheath thickness S* > S0. Here, n0 diagnostic theory. and S0 are the undistorted plasma density and the corre- Indeed, all steady state sheath models are valid for fre- sponding sheath width. quencies much smaller than the ion plasma frequency According to Ref. 21, the plasma density at the probe (x xpi) and are not applicable for frequencies x xpi surface is about one-fifth of the density of the unperturbed where microwave probes operate. For the same sheath struc- plasma. The measured electron temperature is by about 40% ture and size, the sheath capacitance is different for low and less than in the unperturbed plasma. The plasma density high frequencies and is defined not only by the sheath width depletion spreads the distance of an order of magnitude but also by the reaction of ions and electrons on the sheath rf 27,28 larger than the probe radius. These huge plasma disturbances electric field. At x xpi, ions in the sheath are frozen, are typical for collisional plasmas controlled by ambipolar while electrons are not. diffusion considered in Refs. 20 and 21. For lower gas pres- Expression (1) for the floating potential at kD Randa sures, when ion inertia and charge exchange processes domi- Maxwellian EEDF is never valid in gas discharge plasmas for nate plasma transport to the probe, the plasma depletion high energy electrons (e > eVsf) that form the floating poten- 22 around the probe is smaller, but still significant. tial. Depending on the shape of the EEDF, Vsf values can be As far as we know, the effect of plasma depletion considerably smaller or considerably larger than that given by (although mentioned in the same MP works) has not been Eq. (1) with Te ¼ Teff ¼ 2/3hei,wherehei is the electron energy considered in application to microwave probes. Existing pub- averaged over EEDF; see Ref. 29 and Fig. 3 in Ref. 15. lications on microwave probes assume a uniform plasma, The value of the floating potential, Vsf,isdefinedby undistorted by the probe. Plasma depletion also occurs around the equilibrium of the electron and ion currents to the a Langmuir probe (LP), but there is a fundamental difference probe, Ii þ Ie ¼ 0. For non-Maxwellian EEDF, electron cur- between MP and LP. MP senses the sheath and plasma per- rent to a floating probe, Ie, is defined by fast electrons with mittivity within the distance of about a probe radius, R, while their energy, e > eVsf, having distribution temperature, Teh, LP senses electrons coming from the plasma volume with while the ion current, Ii (as well as kD and vs), is defined by radius K or ke (whichever is smaller), where K is the plasma the electron screening temperature, Tes weighted by slow 30,31 size and ke is the electron mean free path. In the validity range electrons. of a Langmuir probe, K Randke R, and the plasma In capacitively coupled plasma (CCP) and inductively 2 depletion effect is of the order of (R/ke) and, therefore, is coupled plasma (ICP) at very low pressures, when collision- negligible. less rf power absorption dominates, EEDFs are usually In all known models for various microwave probes, the enriched with low-energy and high-energy electrons, and thickness of the sheath surrounding a floating MP defines its Teh Tes,Vsf can be estimated as electrical capacitance Cs found as the capacitance of a vac- Vsf ¼ðTeh=2eÞlnðMTeh=2pmTesÞ; (2) uum gap of thickness S ¼ kkD, where kD is the Debye length. The value of k depends on the chosen sheath model and the which may considerably exceed V given by (1) with chosen definition for the plasma-sheath interface. The values sfM T ¼ T . The uncertainty in V together with the uncer- of k found in the literature for a probe under floating poten- e eff sf tainty in k and evaluations for non-Maxwellian EEDF tial range between 3 (Ref. 4) and 14.23 Such a significant D e (that, as a rule, is the case for real gas discharge plasmas) spread in the estimation of the k values causes a significant should bring an additional uncontrollable error in inferring uncertainty in the evaluation of the sheath width. At very plasma parameters from the microwave probe measurement. high negative sheath voltage jV jT /e when S k , the s e D0 The detrimental effect of the plasma rf potential, V on sheath width can be defined by the Childs-Langmuir Law for p, the Langmuir probe sheath is well known. Practical methods ions but becomes inaccurate 13 for typical MW probes under to mitigate this effect in the probe measurement in rf plasmas the sheath floating potential for the Maxwellian electron have been quite well understood.14 To the best of our knowl- energy distribution function, EEDF edge, the effect of the rf plasma potential Vp on the MW

VsfM ¼ Te=2e lnðM=2pmÞð3-6Þ Te=e: (1) probe measurement has never been recognized and addressed in the published experiments. Note that VsfM is somewhat different (about twice) for light Since the microwave probe circuit is grounded for rf and heavy gases, which has never been accounted in the MW plasma sustaining frequency, the presence of the rf plasma probe models. Since there is no strict boundary between the potential would inevitably cause a modulation of the micro- plasma and the sheath, various models are used to describe wave probe sheath with the rf plasma driven frequency. The the sheath. One of them is the Bohm sheath model,24 which rf modulation of the MW probe sheath results in the two 060702-3 Valery Godyak Phys. Plasmas 24, 060702 (2017)

following negative effects that further deteriorate the MW general, s depends on the frequency, geometry, and the par- probe accuracy. ticular shape of EEDF. Moreover, the spatial profile of the First, at Vp Te, that is typical for ICP and to a much microwave field near the probe is actually unknown and is larger extent in CCP, the MW probe modulation would expected to be different for MW probes operated at x < xpe increase the dc negative probe floating potential from Vsf to and those operated at x > xpe. The calculations of micro- 32 Vsfrf given by the following expression: wave field distribution around the MW probe are rather prob- ÂÃ lematic with many existing formulas for different kinds of Vsfrf ¼ Vsf þðTe=eÞln I0ðeVp=TeÞ : (3) skin effects (like collisionless, normal, anomalous, and non- linear), yet derived for a flat plasma boundary. Here, I is the modified Bessel function and V is the ampli- 0 p We have shown that existing microwave probe theories tude of rf plasma potential. Expression (3) is valid for and experiments are built with too simplified and unrealistic R k and Maxwellian EEDF. The increase in the sheath D assumptions about plasma uniformity, sheath capacitance, negative dc voltage results in the increase in the sheath width and absence of rf plasma potential interaction with the probe and in the reduction in its capacitance, which affects the sheath and the assumption of a Maxwellian EEDF. The latter MW probe resonance frequency. For an argon rf plasma with is usually assumed when the EEDF is not known. T ¼ 3 eV and a quite moderate amplitude of V ¼ 20 V, we e p In gas discharge plasmas, EEDFs are non-Maxwellian find from Eq. (3) that the value of V is twice large than sfrf for the energy range related to electron transport (elastic the value of V given by Eq. (1). sf energy range) and excitation, ionization, and electron escape Second, the rf modulation of the probe sheath causes the to the walls (inelastic energy range). Plasma parameters, like modulation of its resonance frequency. Since the period of complex plasma permittivity, electron collision frequency, the rf oscillation is much shorter than the acquisition time of Debye length, ion sound speed, and probe floating potential, the MW probe response, the probe rf modulation results in a are sensitive to the EEDF.30,31,35,36 Assuming a Maxwellian convoluted resonance curve which gives an exaggerated EEDF in the calculation of these parameters in the micro- value for the electron collision frequency, ¼ Dx. e wave probe theory applied to a real plasma would lead to In recent publications on microwave probes, the plasma uncontrollable errors in the inferred plasma parameters. around the MP was described with the complex permittivity In summary, current MP theories have some unrealistic of the cold plasma. In such an approach, the electron colli- assumptions and involve some plasma parameters, which are sion frequency with atoms, , was found to be ¼ Dx,or e e only qualitative estimates and are not accurate enough for a more accurate, from Ref. 5. A kinetic theory of a MW probe reliable plasma diagnostics theory. of arbitrary shape has been presented in Ref. 4, but it is not suitable for practical application in diagnostics with the MW This work was partially supported by the DOE OFES probe. (Contract No. DE-SC0001939). In the majority of publications on microwave probes, the measurements were performed in a low pressure rf 1J.-H. Kim, D.-J. Seong, J.-Y. Lim, and K.-H. Chung, “Plasma frequency plasma where the length of the microwave field localization, measurements for absolute plasma density by means of wave cutoff meth- d, (nearly equal to the MP radius, R), is comparable to or od,” Appl. Phys. Lett. 83, 4725 (2003). 2M. Lapke, T. Mussenbrock, and R. P. Brinkmann, “The multipole reso- less than the electron mean free path, ke տ d R: In this nance probe: A concept for simultaneous determination of plasma density, case, the electron thermal motion in the non-uniform micro- electron temperature, and collision rate in low-pressure plasmas,” Appl. wave field of the MP may cause a change in the plasma dis- Phys. Lett. 93, 051502 (2008). persion relation and give raise to plasma collisionless 3C. Schulz, T. Styrnoll, R. Storch, P. Awakowicz, T. Musch, and I. Rolfes, absorption which is well known in low-pressure capacitive “The multipole resonance probe: Progression and evaluation of a process 33,34 տ compatible plasma sensor,” IEEE Sens. J. 14, 3408 (2014). and inductive discharges, when vTe dx, where vTe is 4J. Oberrath and R. P. Brinkmann, “Active plasma resonance spectroscopy: the electron thermal velocity. At the typical MP frequency, A kinetic functional analytic description,” Plasma Sources Sci. Technol. 10 1 x 10 s , the condition vTe տ dx could be satisfied for 23, 045006 (2014). 5 thin MPs (like hairpin, plasma absorption, and cutoff K. H. You, S. J. You, D. W. Kim, B. K. Na, B. H. Seo, J. H. Kim, and H. Y. Chang, “Measurement and analysis of electron-neutral collision fre- probes), and the inferred electron collision frequency could quency in the calibrated cutoff probe,” Phys. Plasmas 23, 033509 (2016). be affected by the collisionless absorption. For microwave 6K. H. You, S. J. You, D. W. Kim, B. K. Na, B. H. Seo, J. H. Kim, D. J. frequency, only the fastest electrons of the electron energy Seong, and H. Y. Chang, “Measurement of electron density using reac- distribution (whose distribution temperature, T , usually tance cutoff probe,” Phys. Plasmas 23, 053515 (2016). eh 7D. W. Kim, S. J. You, J. H. Kwon, K. H. You, B. H. Seo, J. H. Kim, J.-S. strongly deviates from that of the main body of electrons) Yoon, and W. Y. Oh, “Reproducibility of the cutoff probe for the measure- contribute to collisionless power absorption. Therefore, with- ment of electron density,” Phys. Plasmas 23, 063501 (2016). out knowledge of electron energy distribution corresponding 8D. W. Kim, S. J. You, J. H. Kim, H. Y. Chang, and W. Y. Oh, to the EEDF tail, the evaluation of collisionless absorption “Computational comparative study of microwave probes for plasma den- sity measurement,” Plasma Sources Sci. Technol. 25, 035026 (2016). remains problematic. 9J. Oberrath and R. P. Brinkmann, “Influence of kinetic effects on the spec- Accounting for collisionless absorption in MP models trum of a parallel electrode probe,” Plasma Sources Sci. Technol. 25, 065020 (2016). by some combined electron collision frequency, eff ¼ e 10R. L. Stenzel, Rev. Sci. Instrum. 47, 603 (1976). þ with ¼ v /4d gives only a qualitative estimate that is 11 s s Te C. Schulz, T. Styrnoll, R. Storch, P. Awakowicz, and I. Rolfes, “The pla- not suitable for a diagnostic theory. Indeed, s cannot be nar multipole resonance probe: Challenges and prospects of a planar expressed by a simple universal formula like above since, in plasma sensor,” IEEE Trans. Instrum. Meas. 64, 587 (2015). 060702-4 Valery Godyak Phys. Plasmas 24, 060702 (2017)

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