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Physics of Thin Films

PES 449 / PHYS 549

Plasma Physics

 references:

  • "Classical Electromagnetic Radiation" M. A. Heald and J. B. Marion
  • "Foundations of Electromagnetic Theory" J. R. Reitz, F. J. Milford and R. W. Christy
  • "Electromagnetic Fields and Waves" P. Lorrain, D. R. Corson, and R. Lorrain

Plasmas

Plasma:
  • dilute ionized gas
  • (often at high temperature)
  • contains free electrons (light) and positive ions (heavy)
  • excellent conductor with current mainly carried by electrons

Usually we concentrate on the behavior of the electrons because they are more mobile.

Creating a plasma:

If we start with a gas of neutral atoms, we create a plasma by removing an electron from an atom, leaving a positive ion.

Typical ionization energies are shown in the table:

Element
First Ionization Potential
Second Ionization Potential

Argon

15.7 eV

27.76 eV

Mercury

10.3

.

Neon

21.4

.

Oxygen

13.6

34.93

Sodium

5.1

47.0

Chromium

6.7

16.6

Since 1 eV corresponds to about 11,600 K, it is typically not practical to achieve ionization by thermal processes. Instead we rely on electron collisions with atoms.

electrons ionize by collision most effectively for energies around 100 eV

ionization vs. electron energy

element

number of ions formed per cm of travel in 10 mTorr gas for electrons with about 100 eV
He
0.015
Ne
0.025
H2
0.040
N2
0.100
Ar
0.110
Hg
0.210

electron collisions can also produce excited (but not ionized) atoms,

collisions between excited atoms and ground state atoms can lead to ionization of the ground state atom (Penning ionization)

characterizing a plasma:

characterize by temperature (energy), electron density (Ne) and particle density or neutral atom density (N)

 

  • cold plasma
    • particle energy of a few eV
    • typical of most thin film processes
  • hot plasma
    • particle energy of a few thousand eV
    • typical of nuclear fusion and some astrophysics
  • electron temperature often > ion temperature
    • especially in dilute plasmas

     

  • density
    • at pressure of 5 mTorr: total particle density about 1013 particles/cm3
    • weakly ionized: ion density = electron density is about 108 ions/cm3
    • strongly ionized: ion density = electron density is about 1012 ions/cm3

    These parameters are often grouped as follows:

  • Debye length
    • distance over which significant charge separation can occur
    • lD(cm) = 743 (Te / Ne)1/2
      • with density in electrons / cm3
  • plasma frequency
    • to be derived shortly
    • wp = 56,548.67 Ne1/2
      • with density in electrons / cm3
  • critical degree of ionization
    • if Ne / N is much greater than this critical degree of ionization, the plasma behaves as though it is fully ionized
    • ac = 1.73 x 1012 seA Te2
      • where seA is the electron-atom collision cross section in cm2 (typically 10-16 - 10-15)

Electrostatics

  • charged particles in a constant Electric field (E) with no magnetic field (B = 0)

    When subjected to Electric field:

    • charges redistribute themselves to shield the interior from the fields
    • plasma region is field free and approximately neutral
    • for T = 2000 K and N = 1018 electrons/meter3, sheath thickness is about 1 micrometer

      plasma surrounded by sheath

  • electrons in a constant Magnetic Field (B) with no electric field (E=0)

     

    • spiral motion in along B field lines

    equations to Larmor radius

    frequency of gyration = wcyclotron = qB / m

    note: electrons have a longer actual path length before reaching one side of the system => more likely to ionize a neutral gas atom.

  • electrons in a uniform, constant E and B with E perpendicular to B
    • motion has three components
      1. constant velocity vparallel in direction of B
      2. gyration about the B field lines
      3. constant drift velocity vd = E/B perpendicular to E and B

      note vd does not depend on mass or charge - so all particles drift together

      velocity components for plasma in crossed fields

Electrodynamics

apply an oscillator model to electrons in the plasma
  • at low frequencies (<50 kHz) ions and electrons both oscillate
  • at high frequencies (>50 kHz) heavy ions can not follow switching fields => only electrons oscillate while ions are relatively stationary

Examine forces on electrons:

  • driving force from varying E field
  • no restoring force since electrons are not bound (spring constant = 0)
    • not true if charge separation in plasma leads to electrostatic restoring forces
  • damping term, g, from collisions (this could be large)
    • g = collision frequency
  • consider effects of electromagnetic wave on plasma (with no static fields)

from F = ma we can write down an equation of motion:

equations for charges in plasma

Special case: Dilute Plasmas

dilute => few collisions so g is small (<< w)

same approximations as before

equations for a dilute plasma

for g < w < wp (evanescent domain)

k is imaginary => waves are attenuated

 

When is a plasma dilute ?

criteria: g << w ..... to put numbers in, let us say g = 0.01 w

estimate collision frequency from kinetic theory of gasses:

collision frequency equation

ąd2 = seA = electron - atom collision cross section

using typical values

  • d = 1 Å = 1 x 10-10m
  • m = 9.1 x 10-31 kg
  • T = 10 eV = 116,000 K

    then g = 6.65 x 10-14 N (where N is in particles/m3)

    if the incident frequency, w , is 1000 Hz

    • then we want g < 10 Hz which means N < 1.5 x 1014 atoms/m3 (or 1.5 x 108 atoms/cm3 or 4 x 10-9 torr)

    if the incident frequency is 1,000,000 Hz

    • then we want g < 10,000 Hz which means N < 1.5 x 1017 atoms/m3 (or 5 x 1011 atoms/cm3 or 4 x 10-6 torr)

    Reality check: 1 torr = 3.5 x 1022 atoms/m3 . . . . so most plasmas in our processes are NOT dilute

    Another reality check: is the plasma frequency for these conditions greater than the incident frequency ?

    Using N = 1.5 x 1017 atoms/m3 (and assuming f = 0.1 and q = 1.6x 10-19C), plasma frequency = 7 x 109 Hz which is much greater than anything else.

Special Case: Dense Plasmas

collisions between charged particles are common
electrons and ions are in thermal equilibrium

electrons and ions move together

equilibrium theory formulation of plasmas

particles maintain a Maxwell-Boltzmann velocity distribution

kinetic properties and transport properties of particles can be calculated from this.

Applications of plasmas

  • cleaning / etching of surfaces
  • sputter deposition source
  • bombardment during deposition to modify film
  • activation of reactive gasses

Plasma Sources

  • plate electrodes
    • low plasma densities (109 - 1010 charged particles per cm3)
    • common in sputter deposition
    • discuss further during sputter deposition
  • Inductively coupled plasma (ICP)
    • high plasma densities (1011 - 1012 charged particles per cm3)
    • operates well at lower gas densities (< 50 mTorr)
    • can be used up to atmospheric pressures (and beyond)
    • couple RF energy inductively into plasma (lossy electrical conductor)
      • produces more efficient ionization
  • Helicon
    • high plasma densities (1011 - 1012 charged particles per cm3)
    • operates well at lower gas densities (< 50 mTorr)
    • radiates RF energy into plasma for resonant absorption
      • produces more efficient ionization
  • Electron cyclotron resonance (ECR)
    • high plasma densities (1012 - 1013 charged particles per cm3)
    • operates well at lower gas densities (down to 0.1 mTorr)
    • couples microwave energy to electrons by matching frequency to electron gyration frequency
      • wc = eB / me
      • produces more efficient ionization
    • control the plasma density with microwave power and gas pressure
    • can also control ion species created (O2+, O+)

    plasma sources

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