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EE-527: MicroFabrication Vacuum Systems R. B. Darling / EE-527 / Winter 2013 Outline • Vacuum principles • Vacuum pumps • Vacuum systems • Vacuum instrumentation • Vacuum materials and components R. B. Darling / EE-527 / Winter 2013 Uses of Vacuum in Microfabrication Rough Vacuum High Vacuum Ultra-High Vacuum wafer chucks evaporation surface analysis load locks ion implantation molecular beam epitaxy (MBE) sputtering reactive ion etching (RIE) low pressure chemical vapor deposition (LPCVD) R. B. Darling / EE-527 / Winter 2013 Units of Pressure Measurement • 1 atmosphere = – 760 mm Hg = 760 torr – 760,000 millitorr or microns – 29.9213 in. Hg – 14.6959 psi – 1.01325 bar 760 mm Hg – 1013.25 millibar 33.93 ft H2O – 101,325 pascals (Pa) – 407.189 in. H2O – 33.9324 ft. H2O 1 Pascal = 1 N/m2 1 torr = 1 mm Hg 1 micron = 1 m Hg R. B. Darling / EE-527 / Winter 2013 Vacuum Ranges • Low or Rough Vacuum (LV) – 760 to 1 torr • Medium Vacuum (MV) –1 to 10-3 torr • High Vacuum (HV) –10-3 to 10-7 torr • Ultra-High Vacuum (UHV) –10-7 to 10-12 torr R. B. Darling / EE-527 / Winter 2013 Partial Pressures of Gases in Air at STP Gas Symbol Volume Percent Partial Pressure, Torr Nitrogen N2 78 593 Oxygen O2 21 159 Argon Ar 0.93 7.1 Carbon Dioxide CO2 0.03 0.25 Neon Ne 0.0018 1.4 x 10-2 Helium He 0.0005 4.0 x 10-3 Krypton Kr 0.0001 8.7 x 10-4 -4 Hydrogen H2 0.00005 4.0 x 10 Xenon Xe 0.0000087 6.6 x 10-5 Water H2O Variable 5 to 50, typ. Standard Temperature and Pressure (STP): 0C and 1 atm. R. B. Darling / EE-527 / Winter 2013 Ideal Gas Law - 1 • V = volume of enclosure • N = number of molecules • Nm = number of moles = N/NA • n = particle density = N/V • P = pressure • T = absolute temperature -23 • kB = Boltzmann’s constant = 1.381 x 10 J/K 23 • NA = Avogadro’s number = 6.022 x 10 particles/mole • R = Gas constant = NAkB = 8.315 J/mole-K PV Nm RT PV NkBT P nkBT R. B. Darling / EE-527 / Winter 2013 Ideal Gas Law - 2 • Historical Laws for ideal gases: – Boyle’s Law: P1V1 = P2V2 at constant T – Charles’ Law: V1/T1 = V2/T2 at constant P – Gay-Lussac’s Law: V = V0(1 + T/273) – Henry’s Law: Pi = Kixi (partial pressure of a gas above a liquid) – Dalton’s Law: P = P1 + P2 + … + Pn (sum of partial pressures) • Real gases are better described by van der Waals’ equation of state: an2 P V nb nRT 2 V – an2 term accounts for intermolecular attraction – nb term accounts for intermolecular repulsion R. B. Darling / EE-527 / Winter 2013 Kinetic Gas Theory • Velocity of a molecule is v vx xˆ vy yˆ vz zˆ 2 2 2 2 • Mean square velocity is v vx vy vz 2 • Pressure exerted on a wall in the x-direction is Px nmvx 2 2 • If velocities for all directions are distributed uniformly, v 3vx • Thus, 1 2 1 2 3 P 3 nmv nkBT 2 mv 2 kBT • Each molecular DOF has an average excitation of kBT/2. R. B. Darling / EE-527 / Winter 2013 Distribution Functions - 1 • Boltzmann’s postulates for an ideal gas: – The number of molecules with x-components of velocity in the 2 range of vx to vx + dvx is proportional to some function of vx only: dN dN dN vx (v2 ) dv vy (v2 ) dv vz (v2 ) dv N x x N y y N z z – The distribution function for speed v must be the product of the individual and identical distribution functions for each velocity component: dN vx,vy,vz (v2 ) dv dv dv (v2 )(v2 )(v2 ) dv dv dv N x y z x y z x y z R. B. Darling / EE-527 / Winter 2013 Distribution Functions - 2 • A mathematical solution to the above equations has the form of (A and vm are constants): 2 2 2 vx / vm (vx ) Ae • Normalization of the distribution functions: 2 2 dN NAevx / vm dv NA v2 N A (v2 )1/ 2 vx x m m 4 1 4v 2 2 3k T v2 v2 dN ev / vm dv 3 v2 B v 3 2 m N 0 0 vm m 1/ 2 2 2kBT dv dv dv 4v dv vm x y z m R. B. Darling / EE-527 / Winter 2013 Distribution Functions - 3 • Normalized distribution function for a single velocity component (Gaussian): 1/2 1/ 2 2 2kT 2 m mvx B (v ) exp vm x m 2kBT 2kBT • Normalized distribution function for velocity magnitude (Gaussian): 1/2 3/ 2 2 2 m mv 3kTB (v ) exp vrms m 2kBT 2kBT • Normalized distribution function for a randomly directed velocity (Maxwellian): 3/ 2 1/2 m mv2 8kT 2 2 v B (v ) 4 v exp a 2kBT 2kBT m R. B. Darling / EE-527 / Winter 2013 Distribution Functions - 4 Maxwellian Distribution of Randomly Oriented Velocities: (v2) 1/2 8kTB va m This is sometimes referred to as the average thermal velocity. velocity, v 0 va R. B. Darling / EE-527 / Winter 2013 Impingement Rates • The number of molecules with a velocity from vx to vx + dvx 2 is dNvx = N(vx ) dvx. • A = area under consideration. • Only those molecules within striking distance vxdt will hit the wall after dt seconds. • The number of molecules with velocities from vx to vx + dvx impinging upon the wall per time dt is 1/ 2 N k T d 2 N Av (v2 ) dv dt v (v2 ) dv B vx x x x integrate: x x x V 0 2m 1/ 2 dNi N kBT 1/ 2 2mkBT P Adt V 2m R. B. Darling / EE-527 / Winter 2013 Mean Free Path • MFP is the average distance a gas molecule travels before colliding with another gas molecule or the container walls. • It depends only on the gas density and the particle size. • is the diameter of the particles • 2 is the cross-sectional area for hard-sphere collisions V kT MFP B NP2222 For common gases, {H2O, He, CO2, CH4, Ar, O2, N2, H2}, at T = 300 K: 5 x 10-3 torr-cm Mean Free Path (cm) = Pressure (torr) R. B. Darling / EE-527 / Winter 2013 Viscosity of a Gas • Viscous range: λ < D: – The viscosity is independent of gas pressure or density: 1/2 4mk T 0.499nmv 0.499 B a 3/2 2 • Free molecular range: λ > D: 1/2 Pmv m fm P 42kTBB kT • Viscosity units: – 1 Pa-sec = 10 Poise (P); air at RTP has η = 1.78 x 10−4 P. –10−2 Poise = 1 cP; water at 25°C and 1 atm has η = 0.899 cP. R. B. Darling / EE-527 / Winter 2013 Thermal Conductivity of a Gas • Conductive heat flow: 2 QT W/m • Thermal conductivity: [W/m-°K] (in the viscous range) 1 95cc 4 vv • cv = specific heat at constant volume • cp = specific heat at constant pressure • γ = cp / cv. • The thermal conductivity of a gas is directly proportional to the viscosity and is also independent of gas pressure or density, as long as λ < D = chamber size. R. B. Darling / EE-527 / Winter 2013 Surface Accommodation Coefficient • The accommodation coefficient, 0 < α < 1, is a measure of how well a gas molecule equilibrates to a surface’s temperature. • If α = 1, the kinetic energy of the recoiling gas molecule will be equal to that of the surface temperature. • The thermal accommodation coefficient between surfaces 1 and 2 will be 12 1212 • The accommodation coefficient depends upon the gas species and the local roughness of the surface. • Perfectly elastic collisions with a specular surface give a zero accommodation coefficient. R. B. Darling / EE-527 / Winter 2013 Conductive Cooling in a Low Pressure Gas • When the pressure is low enough that λ > D, gas molecules bounce between walls more frequently than they interact with each other. • The free molecular thermal conductivity is then 1/2 1 ccpvv cc pvk B fm 88ccTpv cc pv mT • The conductive heat flow from surface 2 to surface 1, with T2 > T1 is then cc v QPTT pv1 PTT W/m2 21fm 21 21 ccpv 8 T1 R. B. Darling / EE-527 / Winter 2013 Gas Flow - 1 • Viscous Flow – occurs for pressures greater than ~ 10-2 torr – gas molecules constantly collide with one another – collisions with each other are more frequent than wall collisions – gas behaves like a coherent, collective medium; it acts like a fluid – Knudsen number Kn = λ/D < 0.01. • Transition Flow – Knudsen number 0.01 < Kn < 1. • Free Molecular Flow – occurs for pressures less than ~ 10-2 torr – gas molecules travel for large distances between collisions – collisions with walls are more frequent than with each other – gas molecules fly independently of each other – Knudsen number Kn = λ/D > 1. R. B. Darling / EE-527 / Winter 2013 Gas Flow - 2 • Pipe of radius r and length l: • Viscous Flow – Poiseuille’s equation: r 4 r 4 Q C (P P ) P2 P2 (P P )(P P ) vis 2 1 16 l 2 1 16 l 2 1 2 1 • Free Molecular Flow – Knudsen’s equation: 1/ 2 2 r 3 8k T B Q Cmol (P2 P1) (P2 P1) 3 l m • Ohm’s Law for Gas Flow: Q = CP R. B. Darling / EE-527 / Winter 2013 Gas Throughput •Q = PS • P = gas pressure in torr • S = pumping or leaking speed in liters/second (L/s) • Q = gas throughput in torr-liters/second (torr-L/s) – This is the quantity of gas moving through an orifice per unit time.
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