Mean Free Path: the Way a Gas Particle (Or a Film Particle) Can Travel Without a Collision with Another Particle

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Repetition: Physical Deposition Processes

PVD (Physical Vapour Deposition)

Evaporation

Sputtering Diode-system Triode-system Magnetron-system ("balanced/unbalanced") Ion beam-system

Ionplating DC-glow-discharge RF-glow-discharge Magnetron- discharge Arc-discharge

Ion-Cluster-beam

Reactive versions of the above processes

Repetition: Rates and Cooling Rates PVD

Effusion cell, Ion Gun High rate Evaporation electron gun High rate Sputtering magnetron

-2 -1 3 10 10 1 10 10 10 R[nm/s]

29 3 6 2.9·10 2.9·10 R T [K/s] Eparticle =0.1eV

377 4 7 3.77·10 3.77·10 R T [K/s] Eparticle =1eV These extremely high achievable cooling rates show, that PVD processes (apart from being a direct transition from vapor  solid state) often can be considered as non equilibrium processes .

Vacuum Physics

Central Termini:

Mean free path: the way a gas particle (or a film particle) can travel without a collision with another particle.

Impingement rate: number of particles which hit a surface per second and unit area at constant pressure.

Coverage time: time needed for the formation of a full monolayer.

Mean Free Path I Collision of two particles, 1 and 2 with radius r = R/2:

R 1 r 2 

If both particles are considered as points then a collision happens, if particle 1 is located within a disk of the area  = ·R2.  is called the collision cross section .

Mean Free Path II Aparticle moves straight for a distance l through a gas. Within a cylinder of the volume V = l· it will collide with each particle located in V.



The cylinder contains N = n · V particles. For straight movement this is the collision number .

Mean Free Path III One collision happens if N = 1. This yields the mean free path  as:      1nVn1N 1 1 1    n   Rn 2  rn4 2

Macroscopic information: Particle density n, from the ideal gas equation. Microscoic information: Collision cross section  contains energy dependent atom/molecule radii or the general interaction cross sections of the colliding particles. Mean Free Path IV State of movement of the background gas: Energetic Gas particle: coating particle: relative relative movement movement may may not be neglected be neglected

1 1    rn4 2  rn42 2 Mean Free Path - Example 1 N p  2 B TkNVp n   rn4 V B Tk p = 0.1 Pa -23 kB=1,38·10 J/K T = 300K r = 1.5·10-10m Tk  B   rp4 2  23  ]K[300]K/J[1038.1   3   210 ]m[105.1]mJ[1.04  cm6.14 Mean Free Path - Rough Estimate p=5mmPa p = 1 Pa  = 5 mm p = 10-4 Pa  = 50 m

100

He 10

CO ; Ar; N 2 2 1 H O 2 0,1

0,01

1E-3 Mittlere Freie Weglänge Freie Mittlere [m]

1E-4 10-5 10-4 10-3 10-2 10-1 100 101 102 Druck [Pa] Mean Free Path: Scale Consideration CERN – LHC: U = 2·4.3 · = 27 km p=5mmPa 5 ]mm[  ]Pa[p 5 5 ]Pa[p     ]mm[ 107.2 7   7   9 mbar108.1Pa108.1

Within LHC a pressure of approx. 10-9 mbar has to be maintained, to exclude interparticle collisions.

Gas Phase Transport Clausius' law of distance:

 x  exp)0(N)x(N    

This means:

A significant number of collisions happens before the mean free path is reached . Only approx. 37% of the particles reach  without a collision. The mean free path is only a statistical measure.

Gas Phase Transport - Statistics Consider large ensemble of single situations:

dn d1 d2

Calculate the expectation value of free throw distances:

  x  expx      d  0    x  exp  0    Areal Impingement Rate I Initial situation: Gas molecules hit surface

Wanted: number of gas molecules, which hit the unit surface within 1 second.

Areal Impingement Rate II Approach: cylinder with unit top areas, heigth u.

Only particles with a velocity component u in the

direction of e1, which trespass the top cylinder surface reach the surface within unit time.

1     du)u(n1udz Differential areal u  cylinder  particle impingement rate: volume density

Areal Impingement Rate III

    du)u(n1udz Differential areal u  cylinder  particle impingement rate: volume density

  Total areal  du)u(undzZ impingement rate: u  0 0

um 2 Maxwell-distribution  m  Tk2 of one velocity- )u(  e B  Tkm2 component: B Areal impingement rate IV Calculation of the total areal impingement rate:

  du)u(undzZ  u  0 0 um 2 m   m Tk n   B  Tk2 ndueu  B  B  Tk2 B  Tk2 m 0   B Tk m N p p m n   V B Tk m B  Tk2 Areal Impingement Rate V

Z = Z(p,T,m)= p m =  m B  Tk2

-26 m= 5.3·10 kg (O2) -23 kB=1,38.10 J/K T = 300K p = 0.1 Pa p = 10-4 Pa Z = 2.7·1017 s-1cm-2 Z = 2.7·10 14 s-1cm-2

etwa 270 ML/s etwa 0.2 ML/s Areal Impingement Rate - Graphic Types of Vacuum

Name Pressure [Pa] Mean free Coverage time path [mm] O2, 300K [ML/s]

Rough vacuum Atm15·10-552.7·105  2700 Fine vacuum 1 0.1 5 50 2700  270 -5 5 High vacuum (HV) 0.1 10 505·10 270  0.027 -5 -10 5 10 -7 Ultra high vacuum (UHV) 10 10 5·10 5·10 0.027  2.7·10 -10 10 -7 Extreme UHV (XHV) <10 5·10  2.7·10 

5·105 mm ≡ 500 m 5·1010 mm ≡ 50 000 km (!)

Types of Pumps

Gas transporting: + Rotary pump Rough vacuum/fine vacuum + Diffusion pump High vacuum + Turbomolecular pump High vacuum

Gas adsorbing: + Cold traps Fine vacuum + Cryo pumps High vacuum/UHV + Sublimation pumps UHV + Getter pumps UHV reactive gases + Ione getter pumps UHV inert molecules (activation)

Flow Types Flow through a pipe, diameter d:

Laminar/turbulent: Rough vacuum/fine vacuum   d

d Particle collisions  probable, global flow

Molecular: High vacuum, UHV   d

d  Wall collisions probable, no flow Flow Types and Pumping Systems

Efficient in the laminar region: + Gas transporting pumps: Rotary pump Ejector pump + Rotary pumps, except Turbomolecular pumps

Efficient in the molecular region : + Gas transporting pumps: Diffusion pump Turbomolecular pump + Gas adsorbing pumps

Design Criteria for Vacuum Systems

Mean free path : + Choice of pump type + Pump velocity + Dimension of pipe diameters

Areal impingement rate Z: + Coverage times (e. g. surface analytics) + Impurity content in coatings (ratio of impingement rate of the coating particles and of the background gas particles)

Impurities

Sticking ZDes Z ... Ipingement rate 1 Z ... Desorption rate coefficient : Z Des

High sticking coefficient  ≈ 1 (ZDes ≈ 0): Reactive gases: O2 H20 Complex carbohydrates (pump oil)

Low sticking coefficient  << 1 (ZDes ≈ Z): Inert gases: Noble gases N2 CH4 Carbohydrates without reactive groups

Impurities: Example

Coating material: Al, m = 4.5·10-26 kg Rate Al: 10 nm/s = 3 ·1019 At/(m 2s-1) -26 Impurity: O2, m = 5.3 ·10 kg

Sticking coefficient : approx. 1 für Al und O2 Temperture: 300K Wanted: Background gas pressure, at which 1% Oxygen is incorporated unto the coating

Z 1 p m O2 102   O2 Z 103 19 m  Tk2 Al O2 B  Tk2 2 19 m10310p  B  5 Pa1011.1 O2 m O2 Design Criteria: Summary

Mean free path : Influences gas dynamics. Even at rather high pressures (10-2 Pa) the mean free path reaches the dimensions of average deposition chambers .

Areal impingement rate Z: Crucial for coating purity. The pressure of the background gas has to be at least in the medium high vacuum to guarantee sufficient film purity.