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Electron and Ion Currents

This text explains the concepts of electron and ion currents in plasma, including diffusion and drift terms, the formation of sheath regions, and the effects of conducting and insulated surfaces. It also discusses the floating potential and saturation regimes.

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Electron and Ion Currents

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  1. Electron and Ion Currents • From kinetic theory of gases, impingement rates of electrons and ions within a plasma are: • ze = ne (kTe / 2pme)½ • zi = ni (kTi / 2pmi)½ • These are called diffusion currents • Te >> Ti , me << mi , ne = ni • so ze >> zi ze zi

  2. Electron and Ion Currents • For example, • ne = ni = 1010 cm-3 • Te = 23000 K • Ti = 500 K • Then ze = 2.35 x 1017 cm-2s-1 • Je = eze = 37.6 mAcm-2 • zi = 1.28 x 1014 cm-2s-1 • Ji = ezi = 0.0205 mAcm-2 Je = 37.6 mAcm-2 Ji = 0.0205 mAcm-2 Je >> Ji

  3. Steady-State • No net current can flow through an insulator • Negative charge will build-up on the object repelling electrons and attracting ions (drift currents develop) • A steady-state is achieved when the electron and ion currents are equal diffusion currents (initial) Je = eze Insulated object Ji = ezi diffusion + drift currents (steady-state) - - - - - - eGe Insulated object eGi E

  4. Sheath Region • A positive space-charge region is created that is depleted of electrons, leaving predominantly gas atoms and ions (e.g., Ar, Ar+). • This region is called the sheath region and is similar to the depletion region formed in a semiconductor device such as a p-n junction diode - - - - - - ne Insulated object Sheath region

  5. Dark Spaces • The sheath regions are also called “dark spaces” due to their visual appearance • Fewer electrons result in less optical emission from Mahan, colorplate VI.18

  6. Sheath Currents • At steady-state the impingement rates at the surface are: • For electrons, Ge = -meneE – Dene • For ions, Gi = miniE – Dini • = mobility D = diffusion coefficient Drift Term Diffusion Term - - - - - - eGe Insulated object eGi E

  7. Sheath Currents • In 1-D,  = d/dx, giving: • For electrons, Ge = – meneE – Dedne/dx • For ions, Gi = miniE – Didni/dx • Using ni = ne = n at the edge of the plasma sheath and Ge = Gi (steady-state) gives: • – menE – Dedn/dx = minE – Didn/dx • Solving for E gives: • E = [(dn/dx)/n] [ (Di – De) / (mi + me) ] - - - - - - eGe Insulated object eGi E

  8. Sheath Currents • Substituting this expression for E into the ion flux equation gives: • Gi = – Da dni/dx • Da = (miDe + meDi)/(me + mi) • (ambipolar diffusion coefficient) • Since me >> mi, we have • Da = Di + (mi/me)De

  9. Sheath Currents • Next we can use the Einstein relation between mobility and diffusion, D/m = kT/q, to give: • Da = Di (1 + Te / Ti) • Since Te >> Ti, we have • Da = DiTe/Ti • We see that Da >> Di

  10. Sheath Currents • The effect of the electrons is to establish an electric field that pulls the ions and increases it’s effective diffusion from Di (the unaided diffusion at E = 0) to Da • This effect is known as ambipolar diffusion - - - - - - eGe Insulated object eGi E

  11. Sheath Currents • The ion current increases to • Gi~ ni √(kTe/mi) • For example, for ni = 1010 cm-3, Te = 23000 K, and Ar gas, we have • Gi = 2 x 1015 cm-2s-1 • eGi = 0.35 mA/cm2 • The enhanced ion current is much greater than the unaided diffusive flux calculated previously (ezi = 0.0205 mAcm-2) • Gi ~ surface atom density in 1 sec

  12. Growth Rate Example Gi ~ 1 mAcm-2 = 6.2 x 1015 ions s-1cm-2 Y (1 keV Ar+ ions on Al) ~ 1.5 Sputter rate of Al = 9.3 x 1015 atoms cm-2s-1 Surface density of Al = 6.07 x 1022 atoms cm-3 The deposition rate would be 15 Å s-1 = 5.4 mm/hr

  13. Plasma Potential • Since charged particles are abundant in the plasma, it is a fairly good conductor • The plasma is at an equipotential, Vp, called the plasma potential Insulated object Vp ? sheath region plasma body

  14. Floating Potential • An insulating object placed in a plasma will develop a negative charge • A “floating potential develops” (Vf) until steady-state is achieved (Ge = Gi) Insulated object Vp - - - Ge Vf Gi

  15. Floating Potential Insulated object Vp - - - Ge Vf Gi • M-B distribution of energies: • Gi = Ge = ze exp [ -e (Vp – Vf)/kTe ] • Rearranging gives • Vp – Vf = (kTe/e) ln ( ze / Gi ) • = (kTe/2e)ln(mi/2pme) • e.g., if Te = 23000 K, eze = 37.6 mAcm-2, and eGi = 0.35 mAcm-2 as calculated previously then Vp – Vf = 9.3 V

  16. Conducting Surfaces • A conducting surface at the plasma potential draws the diffusion currents Plasma Potential Cathode Vp Va = Vp ze > zi Va eze ezi • A conducting surface at the floating potential draws no net current Floating Potential Cathode Vp eGe Va = Vf Ge = Gi Va eGi

  17. Saturation Regions (Conducting Surfaces) “Ion saturation” regime Va << Vp: all electrons are repelled Vp - - - - - Va eGi= 0.35 mAcm-2 “Electron saturation” regime Va >> Vp: all ions are repelled + + + + + eze = 37.6 mAcm-2 Va Vp

  18. “Diode” Plasma • Since the electron current is much greater than the ion current, an I-V curve of a conducting surface in the plasma shows rectifying behavior • Hence, the term “diode” plasma from Manos, Fig. 18, p. 31

  19. Langmuir Probe • Can measure I-V curve of plasma using a Langmuir probe from Mahan, colorplate VI.18

  20. Langmuir Probe • From the measured I-V curve, can determine : • Floating potential • Plasma potential from Manos, Fig. 18, p. 31

  21. Conducting Surfaces “Electron Retardation” Regime J< = eGi - eze exp [ -e (Vp – Va)/kTe ] from Manos, Fig. 18, p. 31

  22. Langmuir Probe • Electron temperature, • Te ~ e / [ k dln(J)/dV ] • = 47 840 K from Mahan, Fig. VI.7, p. 166

  23. Langmuir Probe • Electron density can be determined from diffusion current: • ne = (eze) / [e(kTe / 2pme)½ ] • = 5.1x109 cm-3 from Mahan, Fig. VI.7, p. 166

  24. Cathode Fall • The sheath region has low conductivity • Most of the applied potential is dropped across the cathode sheath • Cathode fall ~ Va ~ breakdown voltage cathode Vp - cathode fall V = Va

  25. Cathode Fall • The cathode fall is the kinetic energy gained by ions striking the cathode and of secondary electrons entering the plasma (ignoring collisions in the sheath) • Cathode fall ~ 100’s Volts electrons ions cathode Vp - cathode fall V = Va

  26. Energy Distribution of Sputtered Particles • The sputtered particle energies are much greater than thermal energies • This helps in producing conformal films from Powell, Fig. 2.9, p. 33

  27. Sheath Width • What is the width of the sheath region ? Cathode Vp Va sheath region plasma body

  28. Sheath Width • The width of the sheath (depletion region) can be estimated by calculating the potential that results from a test charge placed within the plasma from Manos, Fig. 2, p. 189

  29. Sheath Width • The charge creates a potential, which in free space (no plasma) would be: • Vo(r) = e / (4peor) • r = distance from the test charge

  30. Sheath Width • The potential in the plasma may be determined by solving Poisson’s equation: • 2V(r) = – r(r)/eo • 2 = Laplacian operator

  31. Sheath Width r(r) = local charge density = e [ ni(r) – ne(r) ] Boltzmann’s law: ne(r) = ne exp [ eV(r) / kTe ] ~ ne [ 1 + eV(r) / kTe ] ni(r) ~ ni since ions are too slow to respond relative to the electrons ni, ne = n = plasma density r(r) ~ – (e2n/kTe) V(r)

  32. Debye Length • 2V(r) = - (e2n/eokTe) V(r) • Solving gives, • V (r) = Vo exp (-r/LD) • LD = Debye length • = (eokTe / e2n)½

  33. Debye Shielding • In free-space, Vo(r) = e / (4peor) • In a plasma, V(r) = Vo exp (-r/LD) • The plasma electrons rearrange to shield the potential causing its attenuation with a decay length equal to LD • The plasma is expelled within a region ~ LD (sheath region) Unscreened potential Vo(r) = Q / (4peor) Q Shielded potential V(r) = Voexp(-r/LD) r

  34. Debye Length •  LD = Debye length • = (eokTe/e2n)½ • = 6.93 [ Te(K) / ne (cm-3) ] ½ • = 743 [ Te(eV) / ne(cm-3) ] ½ • For example, for Te = 1 eV, ne = 1010 cm-3, we get LD = 74 mm

  35. Child’s Law • A more exact treatment for a planar surface (cathode) gives: • Ls = (4eo/9eGa)½(2e/mi)¼(Vp-Va)¾ • Substituting Gias determined previously gives: • Ls ~ 0.8 ¾ LD •  = e(Vp - Va)/ kTe • Hence, the sheath thickness is on the order of 10’s of LD (mm’s) • Electrode spacing ~ cm’s

  36. Cathode Fall from Mahan, colorplate VI.18

  37. Plasma Reactions Homogeneous Reactions (occur within the plasma) Heterogeneous Reactions (occur on a surface)

  38. Homogeneous Reactions • Reactions that occur within the plasma • Excitation : • Electrons produce vibrational, rotational, and electronic states leaving the atom or molecule in an excited state • e- + O2 e- + O2* • e- + Ar  e- + Ar* • e- + O  e- + O* • Glow discharge: • O2*  O2 + hn • O*  O + hn

  39. Homogeneous Reactions • Ionization : • Responsible for ion & electron formation which sustains the plasma • Produces ions for sputtering • e- + Ar  Ar+ + 2e- • e- + O2 O2+ + 2e-

  40. Homogeneous Reactions • Dissociation : • A molecule is broken into smaller atomic or molecular fragments (radicals) that are generally much more chemically active than the parent molecule • This is important in reactive sputtering (e.g., reactive ion etching) and plasma-enhanced CVD • e- + O2 O + O + e- • e- + CF4 e- + CF3* + F*

  41. Heterogeneous Reactions • Reactions that occur on the surface • Sputtering • Secondary electron emission • Reactive etching/deposition

  42. Reactive Ion Sputtering (Deposition) • Excited species (particularly radicals) can react with the surface to deposit nitrides and oxides Reactive sputter deposition : From Ohring, p. 126

  43. Reactive Ion Etching (RIE) • A reactive gas (e.g., N2, O2, CF4) is mixed with the inert gas (e.g., Ar) • The reactive gases are broken down in the plasma into ions, fragments, radicals, excited molecules, etc. from Powell, Fig. 3.18, p. 77

  44. RIE • Acceleration of ions across sheath region results in anisotropic etching Wet chemical etching Plasma etching Ion bombardment adapted from Manos, Fig. 8, p. 12

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