Voltage-dependent coherent drift modes and turbulent transition - PowerPoint PPT Presentation

Voltage-dependent coherent drift modes and turbulent transition regimes in small magnetron devices M. Cappelli T. Ito, N. Gascon, and A. Marcovati, C. Young Stanford University

Magnetron Details • Very small (~ 5 mm diameter discharge, 2 mm gap) • Strong radially-inward B (~ 1 T B peak , 0.3 – 5 T B plasma ) • Low pressure (~100 mTorr) • Very strong axial gradients (mm) – expect gradient drifts • Indium-tin-oxide anode (transparent)

Coherent Rotating Modes E x B is is Cou ounter Clo lockwis ise B E x • unambiguous direction confirmed by varying framing rate • the structures rotate in the - E x B direction (retrograde) • coherent (segmented anode confirms current-carrying) • the total discharge current shows no evidence of fluctuations

Instability Controlled by Gap Voltage Movie frames E 0 L n (V) 6 11 16 21 26 31 36 800 m = 3 m = 4 700 m = 5 600 m = 3 Frequency (kHz) 500 400 300 200 m = 4 100 0 255 260 265 270 275 280 285 Gap Voltage (V) m = 5 • discharge gap voltage controls structure fr freq equency and mode ode • within mode: frequency de decreases with increasing voltage (unexpected)! • wavelength of the modes increase ( m decreases) with increasing voltage

“Turbulent” Regimes between Coherent States E 0 L n (V) 6 11 16 21 26 31 36 800 0.02 m = 3 Ch 1 m = 4 Ch 2 700 m = 5 Ch 3 0.01 600 0 AC Signal (a.u.) Frequency (kHz) 500 −0.01 400 −0.02 300 200 −0.03 voltage between 100 modes −0.04 0 1 2 3 4 5 6 Time ( m s) 0 255 260 265 270 275 280 285 Gap Voltage (V) • temporal behavior of oscillations erratic/turbulent between modes • broad range of frequencies • anode segments serve as “probes” for wavelet analysis • wavenumber Nyquist ~ 280 m -1

Three-wave coupling 3-wave mixing in azimuthal waves k 3 k 2 k 1 𝑔 2 1.31 + 𝑔 3 2.16 = 𝑔 5 3.47 𝑁𝐼𝑨 𝑙 2 −130 + 𝑙 3 100 = 𝑙 5 −30 𝑛 −1 • wavelet analysis reveals high frequency quasi-coherent states (~ five) with strong interconnectivity • three-wave coupling satisfying momentum and energy selection rules

Hypothesis for Retrograde Motion • J comparable to Hall thrusters (~0.1A/cm 2 ) • similar densities/smaller length scale • field reversal necessary to restrict diffusion-driven electrons • plasma rotation in local E x B direction • field drives ions towards anode • field reversals predicted in anode region of Hall thrusters • potential well ~ 5 V • causes ions to stream towards anode simulated • region of strong ionization potential showing field • can potentially lead to reversal in reversal ionization (S-H) spoke instabilities Scharfe, M.K., et al Physics of Plasmas , 13 (8), p.083505.

Hypothesis for Retrograde Motion • J comparable to Hall thrusters (~0.1A/cm 2 ) • similar densities/smaller length scale • field reversal necessary to restrict diffusion-driven electrons • plasma rotation in local E x B direction • field drives ions towards anode • field reversals predicted in anode region of Hall thrusters • potential well ~ 5 V • causes ions to stream towards anode • region of strong ionization • can potentially lead to reversal in ionization (S-H) spoke instabilities • anode-streaming ions seen in early LIF data (Meezan et al 2001) Meezan, et al, 2001. Physical Review E , 63 (2), p.026410.

Gradient Drift-wave Theory source term : Governing Equations ionization and diffusive loss along B 𝜖𝑜 𝜖𝑜 𝜖𝑢 + 𝑾 𝐹×𝐶 ∙ 𝛼𝑜 − 2𝑜 𝑾 𝐹×𝐶 + 𝑾 𝐸 ∙ 𝛼𝑚𝑜𝐶 𝑝 = 𝜑 𝐽 𝑜 + 𝐸 𝐵 electron mass and 𝜖𝑨 momentum 𝜖𝑜 𝜖𝑜 ion mass 𝑾 𝐹×𝐶 = − 𝑪 𝑝 𝜖𝑢 + 𝛼 ∙ 𝑜𝒘 = 𝜑 𝐽 𝑜 + 𝐸 𝐵 2 × 𝑭 𝜖𝑨 𝐶 𝑝 𝜖𝒘 𝜖𝑢 + 𝒘 ∙ 𝛼 𝒘 + 𝑟 𝑾 𝐸 = − 𝑪 𝑝 2 × 𝑙𝑈 Ion momentum 𝑁 𝛼𝜚 = 0 𝑓𝑜 𝛼𝑜 𝐶 𝑝 Linear Perturbation – Fourier Analyzed 𝑓 ෨ 𝑓 ෨ 𝑙 𝑧 𝑙 𝑦 𝜚 𝜚 𝑓 ෨ 2 𝑜 ෤ 𝑙 ⊥ 𝜚 Ions 𝑤 𝑦 = ෤ 𝑤 𝑧 = ෤ = 𝜕 − 𝑙 𝑦 𝑤 𝑝𝑦 𝑁 𝜕 − 𝑙 𝑦 𝑤 𝑝𝑦 𝑁 𝜕 − 𝑙 𝑦 𝑤 𝑝𝑦 2 𝑜 𝑝 𝑁 𝜕 ∗ − 𝜕 𝐸 𝑓 ෨ 𝑜 ෤ 𝜚 𝜕 − 𝜕 𝑝 − 𝜕 𝐸 + 𝑗𝜑 ∗ 𝑦 = − 𝑗𝑙 𝑧 𝜚 = − 𝑗𝑙 𝑧 𝑙𝑈 Elec ෨ ෨ = 𝑊 𝑜 ෤ 𝜕 ∗ − 𝜕 𝐸 𝜕 − 𝜕 𝑝 − 𝜕 𝐸 + 𝑗𝜑 ∗ 𝑜 𝑝 𝑙𝑈 𝐶 𝑝 𝐶 𝑝 𝑓𝑜 𝑝 2 𝜕 𝑝 +𝜕 𝐸 2 2 𝜑 ∗ 2 𝑙 ⊥ 2 𝑙 ⊥ 2 𝑙 ⊥ 𝜕 2 − 2𝑙 𝑦 𝑤 𝑝𝑦 + 𝜕 ∗ −𝜕 𝑝 𝜕 + 𝑙 𝑦 𝑤 𝑝𝑦 2 + 𝑑 𝑡 𝑑 𝑡 𝑗𝑑 𝑡 − 𝜕 ∗ −𝜕 𝐸 = 0 𝜕 ∗ −𝜕 𝐸 net loss (diffusion less ionization) anode-streaming ion velocity 𝜑 ∗ = 𝑙 𝑨 2 𝐸 𝐵 − 𝑤 𝐽 (depends on well depth E o L n )

Gradient Drift-wave Theory source term : Governing Equations ionization and diffusive loss along B 𝜖𝑜 𝜖𝑜 𝜖𝑢 + 𝑾 𝐹×𝐶 ∙ 𝛼𝑜 − 2𝑜 𝑾 𝐹×𝐶 + 𝑾 𝐸 ∙ 𝛼𝑚𝑜𝐶 𝑝 = 𝜑 𝐽 𝑜 + 𝐸 𝐵 electron mass and 𝜖𝑨 momentum 𝜖𝑜 𝜖𝑜 ion mass 𝑾 𝐹×𝐶 = − 𝑪 𝑝 𝜖𝑢 + 𝛼 ∙ 𝑜𝒘 = 𝜑 𝐽 𝑜 + 𝐸 𝐵 2 × 𝑭 𝜖𝑨 𝐶 𝑝 𝜖𝒘 𝜖𝑢 + 𝒘 ∙ 𝛼 𝒘 + 𝑟 𝑾 𝐸 = − 𝑪 𝑝 2 × 𝑙𝑈 Ion momentum 𝑁 𝛼𝜚 = 0 𝑓𝑜 𝛼𝑜 𝐶 𝑝 Simon-Hoh (like) Instability but the B-field curvature (drift) term overtakes the density gradient term relieving the 𝑒𝑜 requirement of the usual S-H condition that 𝐹 𝑝 𝑒𝑦 > 0.

Comparison to Experiments Growth Rate Frequency • peak growth rate depends on well depth (voltage) • increased field (voltage) favors lower mode number (as seen in experiments) • expect a hysteresis (also seen in experiments) • within a mode, increasing voltage decreases frequency

Coherent Fluctuations Drive Transport Current falls to zero 𝑜 𝑜 𝑝 ≈ 1 ෥ Τ (so ) Electron Current Density 𝑜 ෨ 𝐾 𝑓 = 𝑆𝑓 𝑓෥ 𝑊 𝑦 = 2 𝜉 ∗ 𝑙 𝑧 𝑜 𝑝 𝑙𝑈 𝑜 ෤ - 𝜕 ∗ − 𝜕 𝐸 𝐶 𝑝 𝑜 𝑝 2 ≈ 1 𝑜 ෤ 2 𝑓𝑜 𝑝 𝑀 𝛼 𝜉 ∗ 𝑜 𝑝 To match the experiment freqencies 𝜉 ∗ = 3 × 10 6 𝑡 −1 𝑀 𝛼 ≈ 10 −3 𝑛 If we further assume: 𝑜 𝑝 = 10 18 𝑛 −3 𝐾 𝑓 ≈ 0.5 𝐵/𝑛 2 2 Model 𝑜 ෤ ≈ 1 𝜉 ∗ = 3 × 10 6 𝑡 −1 𝑜 𝑝 Consistent with 𝑀 𝛼𝐶 = 2 × 10 −3 𝑛 experimental estimates of current density

Summary • small magnetron discharge can generate very coherent plasma oscillations • fluctuations propagate opposite external E x B direction • likely due to presence of field reversal (potential well) driven by strong gradients in plasma density • drift-wave theory describes this behavior fairly well • “turbulence” between modes • evidence of three-wave coupling within this turbulence • transport during the coherent modes consistent with drift theory • S- H “like” with strong curvature drift term • cross-field current uniquely determined by gradient length scale and ionization Acknowledgements: AFOSR

Extra Slides

Linear Perturbation – Fourier Analyzed 𝜕 ∗ − 𝜕 𝐸 𝑓 ෨ 𝑓 ෨ 𝑙 𝑦 𝜚 𝑜 ෤ 𝜚 = 𝑤 𝑦 = ෤ 𝜕 − 𝜕 𝑝 − 𝜕 𝐸 + 𝑗𝜑 ∗ 𝑜 𝑝 𝑙𝑈 𝜕 − 𝑙 𝑦 𝑤 𝑝𝑦 𝑁 𝑓 ෨ 𝑙 𝑧 𝜚 𝑓 ෨ 2 𝑜 ෤ 𝑙 ⊥ 𝜚 𝑤 𝑧 = ෤ = 𝜕 − 𝑙 𝑦 𝑤 𝑝𝑦 𝑁 𝜕 − 𝑙 𝑦 𝑤 𝑝𝑦 2 𝑜 𝑝 𝑁 2 𝜑 ∗ 𝜕 − 𝜕 𝑝 − 𝜕 𝐸 + 𝑗𝜑 ∗ 𝑦 = 𝑙 𝑧 𝑙𝑈 𝑙𝑈 𝑜 ෤ 𝑦 = − 𝑗𝑙 𝑧 𝜚 = − 𝑗𝑙 𝑧 ෩ 𝑜 ෨ ෨ ෨ 𝑜 𝑆𝑓 𝑓 ෥ 𝑊 𝑊 𝜕 ∗ − 𝜕 𝐸 𝜕 ∗ − 𝜕 𝐸 𝑓𝑜 𝑝 𝐶 𝑝 𝑓𝑜 𝑝 𝑜 𝑝 𝐶 𝑝 𝐶 𝑝 2 𝜕 𝑝 +𝜕 𝐸 2 2 𝜑 ∗ 2 𝑙 ⊥ 2 𝑙 ⊥ 2 𝑙 ⊥ 𝜕 2 − 2𝑙 𝑦 𝑤 𝑝𝑦 + 𝜕 ∗ −𝜕 𝐸 𝜕 + 𝑙 𝑦 𝑤 𝑝𝑦 2 + 𝑑 𝑡 𝑑 𝑡 𝑗𝑑 𝑡 − 𝜕 ∗ −𝜕 𝐸 = 0 𝜕 ∗ −𝜕 𝐸 anode-streaming ion velocity net source (ionization and diffusion) (depends on well depth E o L n ) Characteristics Frequencies 2 𝑙 𝑧 2 𝑙 𝑧 𝜕 ∗ = − 𝑑 𝑡 𝜕 𝐸 = − 𝑑 𝑡 𝜕 𝑝 = 𝑙 𝑧 𝑊 𝐹×𝐶 𝜕 𝑗𝑑 𝑀 𝛼𝑜 𝜕 𝑗𝑑 𝑀 𝛼𝐶

Behavior shows strong dependence on ion mass 5 First experiments with Ne, He 4 • Frequency/mode increases with 3 decreasing M! 2 • Ar (m = 3) =200 kHz, [a.u.] 1 • Ne (m = 1) = 500 kHz 0 • He (m = 0) = 10 MHz -1 Ar • Ar (linear), Ne (non-linear), He -2 Ne He (pulsating at ~0.5-1MHz) -3 0 1 2 3 4 5 6 7 8 9 10 t [  s] Primary scaling of frequency • frequency of oscillations scaling: 1/2 1 ∅ < 𝑔 < ∅ 𝑁 𝑁 • Primary scaling however does not describe the inverse voltage dependence within modes

Complex regimes between azimuthal modes Mode a Mode b coherent coherent m = 3 m = 4 Arrows indicate direction of voltage change (hysteresis) Frequency (kHz) Argon 150 mTorr m=4 m=3 Start Gap Voltage (V) Mode c Mode d Turbulent Turbulent (between m = 3/4) (between m = 4/5)