Plasma and Dry Micro/Nanotechnologies 2. Gas Kinetics Lenka Zajíčková Faculty of Science, Masaryk University, Brno & Central European Institute of Technology - CEITEC lenkaz@physics.muni.cz spring semester 2023 Central European Institute of Technology BRNO I CZECH REPUBLIC Plasma & Dry Technologies . Gas Kinetics ka Zajíčková 2/26 • 2. Gas Kinetics 2.1 Vapors and Gases 2.2 Maxwell-Boltzmann Distribution 2.3 Ideal-Gas Law 2.4 Knudsen Equation 2.5 Energy Stored in Gases 2.6 Units of Measurement 2.7 Mean Free Path 2.8 Knudsen number 2.9 Transport Properties ► p-V-T diagram, critical point, vapors versus gases, saturation vapor pressure ► Maxwell-Boltzmann distribution, mean speed, ideal gas law, Knudsen equation ► energy stored in molecules, equipartition theorem, heat capacity at constant volume and pressure ► standard conditions, units of seem ► mean free path, molecular versus fluid flow (Knudsen number) ► transport of mass (diffusion), momentum (viscous shear) and energy (heat conduction) Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 3 / 26 The possible equilibrium states can be represented in pressure-volume-temperature (p-V-T) space for fixed amount of material (e.g. 1 mol = 6.02 x 1023). Lines = cuts through the p-V-T surface for fixed T => relationship between p and Vm (molar volume). Line a - b - c below the critical point (at T2): ► point a: highest V (lowest p) - vapor phase ► from point a to b: reducing V ->> increasing p ► point b: condensation begins ► from point b to c: V is decreasing at fixed p {b — c line is J_ to the p-T plane, p is called saturation vapor pressure pv or just vapor pressure) ► point c: condensation completed If V is abruptly decreased in b — c transition p would be pushed above the line b - c =>- non-equilibrium supersaturated vapors. Supersaturation is an important drivign force in the nucleation and growth of thin films. Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 4/26 p-V-T diagram It is important to distinguish between the behaviors of vapors and gases: Solid and liquid Constant-temperature line ► vapors: can be condensed to liquid or solid by compression at fixed T, i. e. below critical point defined bypc, Vc and Tc ► gases: monotonical decrease of V upon compression =^ no distinction between the two phases; or sometimes a vapor that is not condensable under conditions encountered in thin-film deposition: P ^ Patmi T > 7~roo m Surfaces "liquid-vapor", "solid-vapor" and "solid-liquid" are perpendicular to the p-T plane =^ their projection on that olane are lines. p (atm) / ^—Equilibrium ^—Equilibrium solid melting -7-> freezing \ CRITICAL POINT vaporization deposition TRIPLE POINT T (°C) Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 5/26 p-T diagram t rej ► triple point: from triple line _L to p-T plane ► below T of triple point: liquid-phase region vanishes condensation directly to the solid phase, vaporization in this region is sublimation ► pressure along borders of vapor region is vapor pressure pv t (°c) ► vapor pressure pv increases exponentially with T up to pc ► pc is well above 1 atm =>• deposition of thin films is performed at p < pc, either p > pv (supersaturated vapors) or p < pv ► first two steps in the deposition (source supply and transport to substrate) should be carried out at p < pv to avoid condensation ► condensation should be avoided also during compression in vacuum pumps Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 7/26 2.2 Maxwell-Boltzmann Distribution Distribution of random velocities V in equilibrium state f(V) = n m 3/2 2tt/cb T exp mV2 2ÍŠTŤ (1) where kB = 1.38 x 10 -23 m2 kg s-2 K_1 (or J K_1) is the Boltzmann constant, n, T and m are particle density, temperature and mass, respectively. If the drift velocity is zero we do not need to distinguish between the velocity and random velocity, i.e. v = V. Maxwell-Boltzmann distribution is isotropic =^ F(v) distribution of speeds v = \ v\ can be defined by integration of f(v) in spherical coordinates Í' TT r2.u F(v)dv= / / f{v)v2sm6d^d0dv Jo Jo (2) resulting in F(v) = 4ttv2 n m 3/2 2tt/cb T exp mV1 2k^Ť (3) Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 8/26 Mean (Average) Speed, Molecular Impingement Flux Vp = Most Probable Speed vav= Average Speed' ■ Vyjns- Root-Mean-Square Speed v_ Speed vp=W -HU Root-mean-square (rms) speed: ^rms — \ I 3kBT m (6) Mean speed: (v) = yav = 1 j F(v)vdv \j 8kBT urn (4) or l/av = SRT W (5) using molar mass M = iwNa in kg/mol and gas constant R kBNA 8.31 Jmol-^-1 where NA = 6.02 x 1023 mol-1 is Avogadro's number The most probable speed vl dF{v) dv V=Vr 2kBT m (7) Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 9/26 2.3 Ideal-Gas Law From the definition of pressure for ideal gas (not necessary to consider pressure tensor but only scalar pressure) 1 3 p = -mn{V2 +V*+ \/f) = ^mn{V2) = m \ V2f(V)d3V. (8) v The ideal-gas law is obtained by integration of (8) using Maxwell-Boltzmann distribution: p = nkB T or pV T = NkB (9) where N is the number of particles. Chemists are used to work in molar amounts (A/a = 6.022 x 1023 mol-1): ► molar concentration nm = n/NA =^ p = nmfl7~ ► number of moles A/m = N/NA p = NmRT/V ► molar volume l/„, = V/Na =>• p = RT/V^ m The ideal gas is obeyed if ► the volume of molecules in the gas is much smaller than the volume of the gas ► the cohesive forces between the molecules can be neglected. Both assumptions are fulfilled for low n =^ always fulfilled for thin film deposition from the vapor phase (T > Troorn and p < patm), i.e. well away from the critical point (most materials pc > 1 atm or if not Tc < 25 °C) Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 10/26 2.4 Knudsen Equation The molecular impingement flux at a surface is a fundamental determinant of film deposition rate: POO PIT/2 r'2.71 V = n(vcos0} = / / /\v)v3 cos(9sin Od^dOdv Jo Jo Jo Substituting Maxwell-Boltzmann distribution (10) fkBT\'/2 1 (11) and using ideal gas law r = p 1 1/2 ZizkTm = pNt 1 1/2 A 2irRTM (12) where M = mNA and R = kNA {M is molar mass) Calculate molecular impinging flux for C02 molecules (44 a. u., 330 pm), 25 °C, 10-3 Pa. Considering the molecule diameter of 330 pm calculate monolayer deposition rate considering all impinging molecules stick to the surface. Plasma & Dry Technologies . Gas Kinetics ka Zajíčková 11/26 2.5 Energy Forms Stored by Molecules Molecules can store energy in various forms. Their energetic states are quantized (spacing between energy levels AE) ► electronic excitations - AEe is highest, transitions between different electronic states are possible only for extremely high T or collision with energetic particle ► vibrational excitations - energy levels correspond to different vibration modes of the molecule, AEV « 0.1 eV (1 eV = 11 600 K) ► rotational excitations - different rotational modes of the molecule, AEr « 0.01 eV ► translational energy - above performed description of molecular random motion Et = 1 /2mV2, no details of inner molecule structure are considered, AEt negligible at ordinary T. E1 'I- Energy level diagram }R equipartition theorem of classical statistical mechanics. Classical statistical treatment assumes very close quantized energy levels of molecules, i.e. approximated as a continuum. It is a good assumption for translational energy when T > 0 K. ► For atomic gases, Et is total kinetic energy content. ► For molecular gases, Er is added at ordinary T and Ev at very high 7~: Molar heat capacity at constant volume Cy (for molecular gas) [J/(mol.K)] - increase of total kinetic energy for increasing T: (13) dT A/A d(£t + Er + Ey) dT A/A (14) for atomic gases for small diatomic molecules at room T 3-R=3-2 2 - two rotational degrees of freedom are excited but vibrational ones are not Plasma & Dry Technologies . Gas Kinetics ka Zajíčková 13/26 2.5 Energy Content of Gas - at constant pressure The heat capacity of any gas is larger when measured at constant pressure cp - heat input is doing pdV work on the surroundings in addition to adding kinetic energy to the molecules: Cp = Cy + R (15) We can write from thermodynamics m dT v (16) where L/m is internal energy per mol L/m = HmA/A ^md cn = m dT (17) where /-/m is enthalpy per mol /-/m = L/m + pV, m m dT dilm\ , / ö Vm + p 07" 07" (18) giving cp = Cy + R Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 14/26 2.6 Units of Measurement SI units?! 1 Torr = 133 Pa = 1 mm Hg 1 bar = 750 Torr = 1.0 x 105 Pa = 0.99 atm (standard atmosphere) The "standard" conditions of T and p (stp) are 0°C and 1 atm (760 Torr). From ideal gas law at stp Vm = 22400 cm3. These conditions are different from standard conditions to which thermodynamic data are referenced: 25°C and 1 bar. In gas supply monitoring - the term "mass" flow rate measured in standard cm3 per minute (second or liters per minute): seem, sees, slm. Standard means 0°C and 1 atm. Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 15/26 2.7 Mean Free Path (M ion or molecule Unless T is extremely high, p is the main determinant of /, / « 1 /p. Mean free path A = crm/7c (19) ► electrons travelling through gas: electrons are much smaller than molecules collision cross section crm is just projected area of the gas molecule A. = 1 /4a2nĚ 7T (20) It's approximation, am is function of el. energy ► ions travelling through gas: similar diameter 1 7ra2A?g (21) ► molecule-molecule collisions: "target" particles are not steady (comparable velocities) mean speed of mutual approach is V2vav rather than yav (on average they approach each other at 90°) it shortens / by V2. Am — 1 V27ra2A?g (22) Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 16/26 It is worth remembering that the mean free path at 1 Pa and room T is about 1 cm for small molecules. The order of magnitude of / is very important in film deposition, because it determines whether the process is operating in the high-vacuum or the fluid-flow regime. The regime is determined by the Knudsen number: Kn = X/L (23) where L is a characteristic dimension in the process, e. g. distance between the source and the substrate, A is the mean free path. ► For Kn > 1, the process is in high-vacuum regime (molecular flow regime). ► For Kn 1. .s * fí- . ____ ..j2-,1 « a . ... • ) mm* ß- ransport Properties - ( netics verview ka Zajíčková 18/26 Proportionality factor Transported quantity Describing equation Derivation from elementary kinetic theory Typical value at 300 K, 1 atm Mass Diffusing flux = 4-m - m^) (Fick's law) Diffusivity = t7/40- + -LY/2 n (cm2} ^MA MB^ DAB s =4c/oe 2 Ar-Ar: 0.19cm2/s Ar-He: 0.72 Momentum Shear stress = x(N/m ) - n^ Viscosity = . ,t i -j VmT Ti(Poise) = -nmcl « 2 a At: 2.26x10^ Poiset He: 2.02x10^ Energy (heat) Conductive heat flux = (Fourier's law) Thermal conductivity = At: 0.176 mW/cm-K He: 1.52 Charge Current density = .( A\ -1 dV dV (Ohm's law) Table 2.1 Gas transport properties from the book by Donald L. Smith, Thin-Film Deposition Principles & Practice, McGraw-Hill 1995. Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 19/26 2.8.1 Diffusion Molecular diffusion is demonstrated using mixture of molecules A (black) and B (white). Consider, the concentration of black molecules A is decreasing from r?a to /?a — Ar?a in the x-direction over a distance of one mean free path A Diffusion of A occurs in the direction of decreasing nA. A rough estimate of diffusion flux can be made by calculating the net flux through an imaginary slab of thickness A (/ in D. Smith book), using ľ = -n{v) where (v) SkTB urn for the fluxes in opposite directions and t => rA = l~(x) - + A) = 1 /4AnA(v) Since An a = A we have rA = -Uv)\%£ I"a = —Dab dnA dx Fik's law, Dab diffusion coefficient of A through B Inserting expression for (v) and molecule-molecule mean free path A = ^a2n we ^'nc' jS/2 Dab m a2 p Empirically, it should be 7"7/4, and m and a are averaged to account for A-B mixture, see next slide. Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 20/26 rn ( Hz jb<4 P*b 4*? o J f01 *wUtt Jgr***-* sxUmju to \l*Lt vneé^ó^ Afr cbtUi^uaJ^ Cxen^y? 0*J ctecv*l Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 23/26 2.8.3 Heat transfer Gaseous heat conduction occurs by transfer of energy in molecular collisions downward along a gradient in molecular kinetic energy Em. Equation for the heat flux O is analogous to that for momentum transfer, with mAu replaced by 2AEr -m 1 dEm O = -n(v)2X- 4 w dx with the units J/(s.m2) or W/m2. We can substitute T for Em using Eq. (14) for molar heat capacity at constant volume (24) dE m dx dEm d T Cy dľ d7" dx NA dx (25) Thus, we obtain Fourier's law O = -K] dT dx (26) where thermal conducitivty KT ~ y/T/mcy/a2. Small, light molecules generally have higher KT although this trend is sometimes reversed by the higher Cy of more complex molecules, which have more rot. and vibr. modes of energy storage. Like r], KT is independent of p for the same reason: as p I, molecular flux r I but A t-However, the situation changes for so low p that the gas is in molecular regime (see next slides). Plasma & Dry Technologies . Gas Kinetics ka Zajíčková 24/26 2.8.3 Heat transfer - molecular regime Heat transfer by gas conduction between two parallel plates: Ts ... common situation in the film deposition, where one plate is a heated platform at temperature Th, and the other is a substrate being raised to 7"s by the heat transfer from the platform. For the gap distance b, the Knudsen number is Kn = X/b. o TrhV ZEZZBZi/JjLU °Íoo°o°0o o°oJo°o° ZZZZZZZ \ Tr8 O - i (a)Kn«l Th (b) Kn>l At the higher p where Kn < 1 (fluid flow), the heat flux is (using KT ~ y/T/mcy/gP) 0 = -/CT— = -^(7h-7s) (27) dx b For Kn > 1 (molecular flow), gas molecules are bouncing back and forth from plate to plate without encountering any collisions =^ use of KT (bulk fluid property) is no longer appropriate. Instead, the heat flux between the plates is proportional to the flux of molecules across the gap (l~) times the heat carried per molecule (using Eq. (25)): O = ry^^ " Ta) = hc(Th - 7S) (28) A/a where Y is the thermal accommodation factor (^ unity except for He) and hc is the heat transfer coefficient (WK/m2) given as hc = yjNA/(2nR) p/\frnT^ Cy Note that hc appears, rather than KT, whenever heat transfer is taking place across an interface rather than through a bulk fluid or other material. Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 25/26 2.8.3 Heat transfer - heat transfer to substrate There are two fundamental difference between the heat flux in the case of fluid Kn 1 regimes: ► O is inversely proportional to b for fluid whereas independent of jb for molecular regime ► O is independent of p for fluid whereas proportional to p for molecular regime One important conclusion which can be drawn for low p is that the heat transfer to a substrate from a platform can be increased by increasing p, but only if the gap is kept small enough that Kn > 1. Helium is often chosen to improve the heat transfer because of its high KT, but in fact it is not the best choice when Kn > 1 because of the thermal accommodation factor in Eq. (28) - for discussion of 7' see the next slide. From Eq. (28), the best choice for a heat-transfer gas is one having low molecular mass to give high l~, while also having many rotational modes to give high Cy. Choices will usually be limited by process chemistry. ■ (N S C CD • l-< u CU 10 O O eS OD tí C3 Eh 10 CD 01 nj O 10 4 \ 1 1 1 1 • • ' I 1—1 >.\ •«X \^-Kn=l ■ > 1 i 1 r f" : iooopa / \ "v \ Y --Kn«l ľ 100Pa **» N. : 10Pa **• X -1_1_1—i_l_1_■ ■ ■ ' .1. 1 1 1 1 1 1 1 1 10' 6 810-3 MT Gap, b (cm) 10" Plasma & Dry Technologies 2. Gas Kinetics Lenka Zajíčková 26/26 2.8.3 Heat transfer - thermal accommodation coefficient In molecular flow regime Kn > 1 (right figure), consider the molecule approaching the heated platform. t/ iitáJtJáIL14 id2Á Oj^O o° O ° n * 0 °0 .°0°.0° L * i v//>J J >/tß*J^^ **>/>/*}>/>/*/>/r/K (a)Kn«l (6) Kn>l It has the temperature 7"rs acquired when reflected from the substrate. Upon being reflected from the platform, it will have temperature 7"rh- The thermal accommodation coefficient 7 is defined as 7"rs — Tvh 7 = T _T ■ (29) ' rs * h It represents the degree to which the molecule accommodates itself to the temperature Th of the surface from which it is reflected. For most molecule-surface combinations, 7 is close to unity, but for He it is 0.1-0.4, depending on the surface. If 7 is less than unity and is the same at both surfaces, the overall reduction in the heat flux represented by 7' is 7 7 (30)