Physique des plasmas radiofréquence Pascal Chabert LPP, Ecole Polytechnique pascal.chabert@lpp.polytechnique.fr
Planning trois cours : Lundi 30 Janvier: Rappels de physique des plasmas froids Lundi 6 Février: Modèle d une décharge capacitive Lundi 13 Février: Décharges capacitives multifréquence
En parallèle: travail personnel ou en binôme Pour le 6 Février: Lecture des chapitres 2 & 3 Pour le 20 Février : Ecrire un modèle global d une décharge inductive utilisée comme source d ionisation d un propulseur à grille
Propulseur à grille excité par une décharge inductive (ICP) Ar Ar e Ar Ar + e Ar Ar + Ar Ar + e e Ar Ar + Ar Ar + Ar Ar + Ar Ar + Ar Ar + Ar
Introduction to plasma discharges 1. Temperature and density domains 2. Non equilibrium 3. Thermal equilibrium properties 4. Collisions and reactions 5. Sheaths, Debye length, Child law
Plasmas Plasma : ionized gas Fully (fusion energy program) partially, ionization degree: x iz Three types of species : n g n electrons Ions (positive and negative) neutrals (radicals or stables) i n i Discharges : x iz 4 1 typically 10 ou 10 5
Plasma discharges ~ Positive space charge: sheaths Plasma Potential is maximum at the plasma center: Electrons are confined Negative ions are confined Positive ions are accelerated toward the walls <V> n + = n e + n - Positive ion flux : directed Electron flux : isotropic Neutral flux : isotropic J i J e J N 0
Relative densities and energies n (cm -3 ) n (cm -3 )
Non Equilibrium Plasma is sustained by electrons which provide ionization e + Ar Ar + + e + e Electrons are usually near thermal equilibrium and have a large average energy: T e is typically 35 000 K Ions and neutrals (heavy) remain near room temperature: 300K However, in discharges, ions gain directed energy toward the surfaces, i.e. they are not in thermal equilibrium
Distribution functions Thermal Equilibrium Collisions, cross sections Mean free path, collision frequency Rate coefficient
Thermal equilibrium properties (1) see page 22
Thermal equilibrium properties (2) see page 24
Thermal equilibrium properties (3) see page 24-25
Averages over Maxwellian distributions (1) see page 23-24
Averages over Maxwellian distributions (2)
Averages over Maxwellian distributions (3)
Averages over Maxwellian distributions (4)
Collisions, cross sections area of targets (cross section): Number of targets in V: Proportion of scattered flux:
Mean free path, collision frequency The flux decays exponentially: The characteristic length is called the mean free path: The collision frequency is : The rate coefficient for the collision process is :
In plasmas The cross section is a function of the incident electron energy Electrons have a velocity distribution. The collision frequency is defined as follows:
Elastic Collisions
Inelastic Collisions, e.g. ionization f ( ) Idealized cross section: ( ) Integral over a Maxwellian yields: E iz Which may be further simplified :
Rate coefficients for reactions K iz e + Ar Ar + + e + e
Plasma Dynamics 1. The central problem of discharge modelling 2. Fluid equations 3. Particle and energy balance 4. Electromagnetic properties waves 5. Radiofrequency Reactors
Central problem of discharge modeling Electromagnetic fields generate forces on particles But, particle motion generates electromagnetic field! To find self-consistent solution Need to solve simultaneously plasma transport and Maxwell s equations Difficult problem; needs simplification Various level of simplification of plasma transport Go from EM fields to Voltage and Currents: circuit theory
Kinetic description Follow the motion of each particle in the field: impossible Define macro-particle and solve the motion of each of these self-consistently with the fields : Particle-in Cell simulations Or, define a distribution function and follow the evolution using Boltzmann or Vlasov equations: kinetic theory All of these are complicated and simpler approaches are often possible
Fluid equations Particle conservation equation Momentum conservation equation
Global model: particle balance Plasma (volume V) Surrounded by a surface A
Global model: energy balance Plasma (volume V) Surrounded by a surface A
Electromagnetic properties The plasma may be treated as a dielectric with the following dielectric constant (page 47): If one ignores displacement currents then the plasma conductivity is:
Dispersion relation of EM waves The plasma is in fact a conductor at low frequency and a dielectric at high frequency
Skin depth Conductor at low frequency ( < pe) Dielectric at high frequency ( > pe) MHz pi rf domain pe GHz Waves are absorbed in a skin depth Propagating waves (microwave diagnostics: interferometry, reflectometry etc.) Inertial (low pressure)
Typical etching reactors: CCP s, ICP s MHz rf domain GHz Electrons follow the rf field pi 13.56 MHz or higher? pe Ions follow time-averaged field ~ rf Capacitively-coupled plasma Inductively-coupled plasma ~ rf ~ rf
Magnetic confinement Anisotropic dielectric constant Cyclotron frequency: c qb m For typical conditions (B 50 Gauss): Larmor radius: R L mv qb Non-magnetized ions: R L Magnetized electrons: R L 10-20cm 1-2 mm
Waves in magnetized plasmas see page 265
Helicon reactors Helicon antenna Source solenoid 400l/s turbo pump Matching network and source cooling rf 13.56 MHz B 0 Ar, SF 6 Helicons generate high density plasmas. Interesting for: Very deep etching Load lock and cartridge transfer Wafer holder Chamber solenoid Space plasma propulsion 150l/s turbo Water cooling
Bounded Plasmas 1. DC sheaths 2. Plasma/sheath transition 3. Plasma transport 4. Plasma flux leaving the plasma and reaching the surfaces
Why sheaths? Without sheaths, currents at the wall are : Positive space charge: sheaths J e en 0 kte 2 m e J i en 0 kti 2 M ~ Plasma n + = n e + n - Since m e << M and T e >> T i : J e >> J i loss of electrons <V> The positive space charge builds up an E field directed to the walls which confines electrons and accelerate ions to the wall 0 E
Debye length (1) Negative potential perturbation Field or potential screening occurs within the Debye length Boltzmann electrons:
Space charge density: Debye length (2) Poisson s equation:
Child law sheath (1) At high voltage, no electrons in the sheath No ion-neutral collisions Positive ion current is limited by the charge space 0 x Plasma J i s -V 0
Child law sheath (2)
Child law sheath (3) Integrate twice over x: 0 x Plasma J i s -V 0
Sheath thickness Positive ion current produced by the plasma (described later in the course): Using current continuity and Child law, 4 9 0 2 q M i J 1 2 h V0 s 3 2 2 e n i l e0 h l kt M en e e0 kt M e + + e - + V 02 >V 01 Plasma + + + + + + + + e - e - V 01 + we obtain the sheath thickness: s ev k 0 De T e 3 4 - I i
Plasma/sheath transition This velocity is called the Bohm velocity and is noted u s = u B The flux at the wall may then be written:
Plasma transport Forces must balance! Electric force Pressure force Friction force
Plasma transport n 0 n s The transport of the plasma, and consequently the ratio h l that controls the plasma flux at the boundary, depends upon the pressure regime: - So-called Schottky or ambipolar diffusion at high pressure - Godyak solution at intermediate pressure regime
Ambipolar diffusion (1) Page 84
Ambipolar diffusion (2) Page 86
Godyak s solution for intermediate pressures Page 87 At low pressure the ion-neutral collision frequency, and consequently the ion mobility, becomes a function of the fluid velocity: This leads to the following edge-to-center density ratio:
To summarize, at low and intermediate pressure n n 0 n s d Plasma Sheath 0 x s x
Sheath at high pressure n n 0 Plasma n s 0 x s x
h l vs pressure
The issue of electronegative plasmas and neutral depletion In the previous theories, we considered only positive ions and electrons, n e =n i, and we considered constant neutral density. However, processing gases are electronegative and the plasma may contains a large amount of negative ions, n e +n n =n i Moreover, contemporary reactors have high plasma densities which may lead to neutral depletion at the reactor center These issues are very important! All transport theories must be revisited (some of these issues will be treated later in this course) Later in this course, we will see the effect of electronegativity on the plasma stability