Vacuum and Thin-Film Deposition from First Principles

Every thin-film lab has the same machine somewhere: a glass or steel bell jar on a steel base plate, a cabinet of pumps underneath humming and occasionally gurgling, and a front panel full of labels nobody explained to you — RV, BV, HV, GH1, GH2, LT, HT, DTM. Most of us learn to drive it by watching a senior, writing the steps down, and hoping. This article is the explanation I wish I had been given: why every step exists, derived from the physics, so that the sequence becomes obvious instead of something to memorise.

It is written for someone who has never touched a vacuum system. If you already have, skip to the simulator or the DTM section.

01The family of deposition techniques

A thin film is a layer of material from a few atoms to a few micrometres thick on a surface we call the substrate. There are three broad ways to put one there:

Physical vapour deposition (PVD): turn a solid into a vapour by physical means — heat it until it evaporates, or knock atoms off it with ions (sputtering) — and let the vapour condense on the substrate. No chemical reaction is needed. PVD almost always needs vacuum.
Chemical vapour deposition (CVD, ALD): let gases react on the hot substrate so that the solid product stays behind (e.g. SiH₄ → Si + 2H₂). Atomic-layer deposition (ALD) does this one self-limiting monolayer at a time.
Liquid and solution routes: spin coating, dip coating, electroplating, sol–gel. Cheap, usually at atmosphere.

Figure 1

Where everything in this article sits. Thermal and electron-beam evaporation are two ways of making the vapour in PVD. Electron-beam lithography uses electrons too, but it deposits nothing: it draws a pattern in a polymer so that a later deposition lands only where you want it.

02What pressure is: kinetic theory in five equations

The air in a room is about $2.5\times10^{19}$ molecules in every cubic centimetre, each flying at roughly the speed of sound and hitting each other and the walls billions of times a second. Pressure is nothing more than the momentum those molecules deliver to a wall per unit area per unit time. Everything in vacuum technology follows from that picture, so let us build it carefully.

1 · The ideal-gas law, molecule by molecule

A molecule of mass $m$ moving towards a wall with velocity component $v_x$ bounces back with $-v_x$: it hands the wall momentum $2mv_x$. Counting how many such hits arrive per second on an area $A$ and averaging over all molecules gives

$$p = \tfrac{1}{3}\, n\, m \,\overline{v^2}, \qquad \tfrac12 m\overline{v^2} = \tfrac32 k_B T \;\;\Rightarrow\;\; \boxed{\,p = n\,k_B\,T\,}$$

where $p$ is the pressure, $\overline{v^2}$ the mean of the squared molecular speeds, $n$ the number of molecules per unit volume, $T$ the temperature in kelvin and $k_B = 1.381\times10^{-23}\ \mathrm{J\,K^{-1}}$ Boltzmann's constant. Pressure is proportional to the number density. Pumping a chamber from 1000 mbar to $10^{-6}$ mbar removes all but one molecule in a billion — and still leaves $2.5\times10^{10}$ of them in every cubic centimetre.

2 · How fast molecules move

The speeds follow the Maxwell–Boltzmann distribution. Its mean is

$$\bar v = \sqrt{\frac{8 k_B T}{\pi m}} \qquad (\text{N}_2 \text{ at } 20^\circ\text{C}: \ \bar v \approx 470\ \mathrm{m\,s^{-1}})$$

Here $\bar v$ is the mean speed of the molecules, $m$ the mass of one molecule (molar mass divided by Avogadro's number), $T$ the temperature in kelvin and $k_B$ Boltzmann's constant.

Light molecules are fast (H₂: 1760 m/s) and heavy ones slow (Au atoms evaporating at 1400 °C: ≈ 420 m/s). This matters for turbo pumps, which work only on molecules they can outrun.

3 · How often molecules hit a surface

The number of molecules striking unit area per second, the impingement rate, is

$$\Phi = \tfrac14 n \bar v = \frac{p}{\sqrt{2\pi m k_B T}}$$

Here $\Phi$ is the impingement rate (molecules per unit area per second), $n$ the number density of the gas, $\bar v$ the mean speed and $p$ the pressure.

For air at room temperature this is $\Phi \approx 2.9\times10^{20}\, p\,[\mathrm{mbar}]$ molecules cm⁻² s⁻¹. The same equation, read in reverse, is the rate at which a hot metal leaves its own surface — the Hertz–Knudsen equation of section 11.

4 · How far a molecule flies between collisions

Picture a molecule of diameter $d$ sweeping out a cylinder of cross-section $\pi d^2$ as it moves. It collides with every molecule whose centre lies in that cylinder. On average it travels the mean free path

$$\lambda = \frac{k_B T}{\sqrt2\,\pi d^2\, p} \qquad\Rightarrow\qquad \lambda_{\text{air}}\,[\mathrm{cm}] \approx \frac{6.6\times10^{-3}}{p\,[\mathrm{mbar}]}$$

Here $\lambda$ is the mean free path, $d$ the molecular diameter (≈ 0.37 nm for N₂), $p$ the pressure and $k_BT$ the thermal energy.

The $\sqrt2$ appears because the other molecules are moving too. At atmosphere $\lambda$ is 66 nm; at $10^{-2}$ mbar it is 6.6 mm; at $10^{-5}$ mbar it is 6.6 m — longer than any chamber.

5 · Units

Unit	Equals	Who uses it
1 Pa (N/m²)	0.01 mbar = 0.0075 Torr	SI; papers
1 mbar	100 Pa = 0.750 Torr	most European/Indian gauges and this article
1 Torr (≈ 1 mm Hg)	1.333 mbar = 133.3 Pa	US equipment, older literature
1 atm	1013 mbar = 760 Torr = 101 325 Pa	“atmosphere”

03Why we need vacuum

Evaporating gold is easy: heat it. Getting the gold atoms to arrive on your substrate as a clean, dense, well-attached film is the hard part, and there are four separate reasons air prevents it.

Reason 1 — the atoms must fly straight

An evaporated atom leaves the source in a straight line. If it collides with gas molecules on the way it scatters, loses energy, may react, and arrives from a random direction — or not at all. The probability that an atom crosses a distance $L$ without a single collision follows from the same reasoning as radioactive decay:

$$P(\text{no collision}) = e^{-L/\lambda}$$

Here $P$ is the probability that an atom flies a distance $L$ without hitting a gas molecule, and $\lambda$ the mean free path.

For a 30 cm source–substrate distance we want $\lambda \gg 30$ cm. At $10^{-2}$ mbar, $e^{-30/0.66}\approx 10^{-20}$: essentially nothing arrives straight. At $10^{-5}$ mbar, $e^{-30/660} \approx 0.96$: 96 % do. That one exponential is why evaporators work at $10^{-5}$–$10^{-6}$ mbar and not at $10^{-2}$.

Figure 2 · Interactive

Gold atoms crossing a 30 cm chamber. The mean free path is calculated from the pressure and drawn to scale. Gold atoms that fly straight are yellow; any atom that has collided with a gas molecule turns red and wanders. Slide from 10⁻² to 10⁻⁵ mbar and watch the transition from diffusion to a molecular beam.

Reason 2 — a clean surface is dirty within seconds

Every gas molecule that hits the substrate may stick. A full monolayer is about $10^{15}$ molecules per cm², so with a sticking probability near 1 the time to cover a clean surface with one layer of contamination is

$$\tau_{\text{mono}} \approx \frac{10^{15}\ \mathrm{cm^{-2}}}{\Phi} \approx \frac{3\times10^{-6}\ \mathrm{s}}{p\,[\mathrm{mbar}]}$$

Here $\tau_\text{mono}$ is the time to form one monolayer, $10^{15}$ cm⁻² the number of adsorption sites per unit area, $\Phi$ the impingement rate of the residual gas and $p$ its pressure in mbar.

At $10^{-6}$ mbar a monolayer of water and hydrocarbons forms in about 3 s; at $10^{-9}$ mbar in an hour. The film you deposit therefore always competes with the residual gas: the ratio of metal flux to gas flux decides how pure it is. Reactive metals (Ti, Cr, Al) pick up oxygen from water vapour readily — which is partly why a fast deposition rate gives a cleaner film.

Reason 3 — hot things oxidise and burn out

A tungsten boat at 1500 °C in air oxidises in seconds; the molten metal in it would oxidise too. In vacuum both survive.

Reason 4 — plasmas and electron beams need room

Electrons accelerated through 10 kV must not hit gas molecules on the way (they would ionise them and arc). Sputtering and glow-discharge cleaning need the opposite: a controlled low pressure at which a discharge can strike (section 12). Either way, you only get the right number of molecules by first removing almost all of them.

Figure 3 · Interactive

The numbers behind “why vacuum”. Mean free path (gold) and the time to form one contamination monolayer (cyan) against pressure. The Knudsen number $\mathrm{Kn} = \lambda/D$ compares the mean free path with the chamber size $D$ and tells you how the gas flows (next sections).

04Degrees of vacuum and the toolbox

Vacuum spans more than fourteen orders of magnitude, and no single pump or gauge covers it. That one fact explains the entire architecture of a coating unit.

Range	Pressure (mbar)	Mean free path	How gas behaves
Rough (low) vacuum	1013 – 1	66 nm – 66 µm	viscous flow: gas moves like a fluid, molecules push each other
Medium vacuum	1 – 10⁻³	66 µm – 6.6 cm	transition (Knudsen) flow
High vacuum (HV)	10⁻³ – 10⁻⁷	6.6 cm – 660 m	molecular flow: molecules only hit walls
Ultra-high vacuum (UHV)	< 10⁻⁷	> 660 m	molecular; walls (outgassing) dominate

Figure 4

Who works where. Each pump, gauge and process has its own pressure window. A rotary pump cannot reach high vacuum; a diffusion or turbo pump cannot start from atmosphere; a Pirani gauge is blind below 10⁻³ mbar and a Penning gauge must not be run above 10⁻² mbar. So we chain them.

The two flow regimes are the key physical distinction. In viscous flow ($\mathrm{Kn} = \lambda/D \lt 0.01$) molecules collide mostly with each other; a pressure difference pushes the gas like water in a pipe, and a pump simply sucks. In molecular flow ($\mathrm{Kn} \gt 0.5$) a molecule bounces from wall to wall without meeting another molecule; nothing “sucks” any more. A pump can only catch the molecules that happen to wander into its mouth, and keep them from coming back. That is why high-vacuum pumps look nothing like low-vacuum pumps.

05Gas flow: throughput, speed, conductance, pump-down

Throughput and pumping speed

We measure gas quantity as $pV$ (by the ideal-gas law, $pV = Nk_BT$ is proportional to the number of molecules). The throughput $Q$ is the $pV$ flowing past a point per second, in mbar·L/s. A pump's pumping speed $S$ is the volume of gas, measured at its inlet pressure, that it removes per second:

$$Q = p\,S \qquad [\mathrm{mbar\,L\,s^{-1}} = \mathrm{mbar}\times \mathrm{L\,s^{-1}}]$$

Because molecules are conserved, the same throughput passes through every part of a series line. This is the most useful equation in the whole subject: if a diffusion pump takes $Q = 5\times10^{-6}\ \mathrm{mbar}\times 500\ \mathrm{L/s} = 2.5\times10^{-3}$ mbar·L/s from the chamber, the rotary pump behind it must take the same $Q$ out of the foreline. With a rotary speed of 4.5 L/s that sets the foreline pressure at $Q/S \approx 5.6\times10^{-4}$ mbar. The diffusion pump has compressed the gas by a factor of 100 — and if the rotary could not keep the foreline low, the diffusion pump would fail (section 9).

Conductance: pipes are resistors

A pipe or valve passes gas in proportion to the pressure difference across it, $Q = C\,(p_1 - p_2)$, where $p_1$ and $p_2$ are the pressures at its two ends and $C$ is its conductance in L/s — exactly like Ohm's law $I = V/R$ (current $I$, voltage $V$, resistance $R$), with throughput playing the current and pressure difference the voltage. Conductances combine like electrical conductances:

$$\frac{1}{C_\text{series}} = \sum_i \frac{1}{C_i}, \qquad \frac{1}{S_\text{eff}} = \frac{1}{S_\text{pump}} + \frac{1}{C_\text{line}}$$

Here $C_i$ are the conductances of the individual pipes, valves and traps connected in series, $C_\text{series}$ their combined conductance, $S_\text{pump}$ the pumping speed at the pump inlet, $C_\text{line}$ the conductance of the line between pump and chamber, and $S_\text{eff}$ the effective pumping speed actually available at the chamber.

The second equation is the sobering one: a pipe can only make a pump slower. The conductance depends strongly on the flow regime. For a long round tube of diameter $d$ and length $L$ (cm) in air at 20 °C,

$$C_\text{viscous} \approx 135\,\frac{d^4}{L}\,\bar p\ \ [\mathrm{L/s},\ \bar p \text{ in mbar}], \qquad C_\text{molecular} \approx 12.1\,\frac{d^3}{L}\ \ [\mathrm{L/s}], \qquad C_\text{orifice} \approx 11.6\,A\ \ [\mathrm{L/s},\ A \text{ in cm}^2]$$

Here $d$ and $L$ are the tube's inner diameter and length in cm, $\bar p$ the mean of the pressures at its two ends, and $A$ the area of an orifice (a hole in a thin wall). The numerical factors hold for air at 20 °C.

Worked example

A roughing line 2 m long and 2.5 cm in diameter. At a mean pressure of 1 mbar (viscous) $C \approx 135 \times 39/200 \times 1 = 26$ L/s — much more than the 4.5 L/s rotary pump, so the line hardly matters. At $10^{-2}$ mbar (molecular) $C \approx 12.1\times15.6/200 = 0.95$ L/s, and the effective speed at the chamber falls to $1/(1/4.5 + 1/0.95) = 0.78$ L/s. Long, thin roughing lines are why the last decade of rough pumping is slow, and why high-vacuum pumps sit directly under the chamber with an inlet as wide as possible.

The pump-down equation

For a chamber of volume $V$ pumped with speed $S$, the gas inside decreases at the rate it is pumped out, while the walls release gas (outgassing, leaks) at a rate $Q_\text{gas}$:

$$V\frac{dp}{dt} = -S\,p + Q_\text{gas} \quad\Rightarrow\quad p(t) = \Big(p_0 - \frac{Q_\text{gas}}{S}\Big)e^{-St/V} + \frac{Q_\text{gas}}{S}$$

Here $p$ is the chamber pressure at time $t$, $p_0$ the pressure when pumping starts, $V$ the chamber volume, $S$ the pumping speed and $Q_\text{gas}$ the total gas load from outgassing and leaks (mbar·L/s).

Two lessons:

Early on ($Q_\text{gas}$ negligible) pressure falls exponentially with time constant $V/S$. The time to go from $p_0$ to $p_1$ is $t = (V/S)\ln(p_0/p_1)$. A 40 L bell jar pumped at 4.5 L/s from 1013 to 0.05 mbar: $t = 8.9 \times \ln(20\,000) \approx 90$ s in theory, a few minutes in practice.
Eventually the pressure stops at the ultimate pressure $p_\text{ult} = Q_\text{gas}/S$. Below about $10^{-4}$ mbar the gas in a chamber is no longer the air you started with; it is water desorbing from the walls. An unbaked steel chamber after an hour releases roughly $10^{-8}$–$10^{-7}$ mbar·L s⁻¹ per cm² of surface, so 5000 cm² of walls and fixtures pumped at 300 L/s settles in the $10^{-7}$–$10^{-6}$ mbar range. Clean, dry surfaces and short venting times matter more than a bigger pump.

06How the pumps work

There are two families. Gas-transfer pumps move gas from the inlet to an exhaust: rotary vane, Roots, diffusion, turbomolecular. Capture pumps hold the gas inside: cryopumps freeze it, ion pumps bury it. Coating units mostly use the first family, a rotary pump and a diffusion (or turbo) pump working as a team.

6.1 The rotary vane pump — the roughing and backing pump

A cylindrical rotor spins off-centre inside a cylindrical stator, touching it along one line at the top. Two vanes slide in slots in the rotor and are pressed against the stator wall by springs and centrifugal force. As the rotor turns, the space between a vane and the contact line on the inlet side grows, so gas flows in; once the next vane passes the inlet, that pocket is sealed off and carried round; on the exhaust side it shrinks until the gas is compressed above atmospheric pressure and pushes open an oil-sealed exhaust valve.

Figure 5 · Interactive

Rotary vane pump, one stage. The vane lengths are computed exactly for the off-centre geometry. Cyan: the pocket opening to the inlet; violet: a sealed pocket being carried round; red: the pocket being compressed towards the exhaust.

Every pumping cycle is a compression from inlet pressure to ≈1.2 bar. One stage achieves a compression ratio of about $10^5$; two stages in series (a two-stage pump, the usual kind) reach an ultimate pressure of about $10^{-3}$ mbar. The oil does three jobs: it seals the tiny gaps between vanes and stator, lubricates, and carries away the heat of compression.

Gas ballast

When the pump compresses water vapour from the chamber, the vapour's partial pressure rises. If it exceeds the saturated vapour pressure of water at the pump temperature (≈ 70–80 mbar at 40 °C) before the exhaust valve opens, the vapour condenses into the oil. Wet oil has a high vapour pressure, the ultimate pressure rises, and the pump corrodes. The gas ballast valve lets a small, controlled amount of air into the compression stroke so that the exhaust valve opens before the water can condense. You trade a slightly worse ultimate pressure for dry oil. Open it after venting with humid air or pumping solvents; close it for the lowest pressure.

6.2 The diffusion pump — the classic high-vacuum (hi-vac) pump

A diffusion pump has no moving parts. A heater at the bottom boils a special low-vapour-pressure oil (silicone or polyphenyl ether) at around 200 °C. The vapour rises up a central chimney and is thrown outwards and downwards through ring-shaped jets at supersonic speed. Gas molecules wandering in from the chamber (in molecular flow, remember — they arrive by chance) are struck by the much denser jet of heavy oil molecules and acquire a downward velocity. Stage by stage they are compressed towards the bottom, where the foreline leads them to the rotary pump. The oil vapour itself hits the water-cooled walls, condenses and runs back into the boiler.

Figure 6 · Interactive

A three-stage diffusion pump. White molecules come in from the chamber; green ones have been struck by a jet and are being driven down. Press Raise the foreline pressure: the jets can no longer push against the gas below them, the pump stalls, and gas and oil stream back up (red). Turn the heater off and nothing is pumped at all.

The name comes from the gas molecules diffusing into the vapour jet. The important numbers:

Speed scales with the inlet area. The ideal speed of an opening of area $A$ is $11.6\,A$ L/s (cm²), and a good pump captures 30–50 % of the molecules that enter it (the Ho coefficient). A 6-inch (150 mm) pump: $0.4\times11.6\times177 \approx 800$ L/s for air.
Critical backing (foreline) pressure, typically 0.2–0.5 mbar. Above it the bottom jet cannot compress against the gas below, the pump stalls, and hot oil vapour moves upwards into your chamber. This is why the rotary must be connected to the foreline (backing it) whenever the diffusion pump is hot.
Maximum inlet pressure, roughly $10^{-2}$–$10^{-1}$ mbar. Above it the jets are overwhelmed by the sheer amount of gas. This is why the chamber must be roughed before the hi-vac valve opens.
Warm-up of 15–30 minutes to boil the oil; cool-down of similar length before the oil may meet air (hot oil oxidises and cracks into tar).
Backstreaming: a few oil molecules always travel upwards. A water-cooled cold cap above the top jet and a chevron baffle (sometimes cooled with liquid nitrogen) between the pump and the hi-vac valve catch them, at the cost of some speed.

6.3 The turbomolecular pump

A turbo pump does mechanically what the diffusion pump does with oil vapour. A rotor carrying many rows of angled blades spins at 20 000–90 000 rpm, so the blade tips move at 200–600 m/s — comparable to the speeds of gas molecules. A molecule that lands on a moving blade leaves with the blade's velocity added to its own, and the angle of the blades means it is more likely to be sent down into the next row than back up. Fixed stator rows between the rotor rows redirect the molecules into the next rotor row.

Figure 7 · Interactive

Rotor and stator rows, unrolled. Slow the blades down and the pump stops discriminating between up and down; speed them up and more molecules are passed towards the backing side. (A simplified probability model; the real transmission depends on blade angle, spacing and the molecule's speed.)

The compression ratio of each stage grows exponentially with the ratio of blade speed to the molecule's thermal speed, roughly $K \propto \exp\!\big(c\,v_\text{blade}\sqrt{M}\big)$, where $K$ is the compression ratio (outlet pressure ÷ inlet pressure for that gas), $v_\text{blade}$ the blade tip speed, $M$ the molar mass of the gas and $c$ a constant of the pump geometry. For nitrogen a turbo pump reaches $K \sim 10^{8}$–$10^{9}$; for hydrogen, whose molecules are fast, only $10^{3}$–$10^{4}$. A turbo is oil-free in the pumping path, starts in minutes, and does not care which way up it is mounted. It still needs a backing pump: its exhaust must be held below about $10^{-2}$–1 mbar.

Figure 8

Pumping speed against inlet pressure (illustrative curves). The rotary pump's speed collapses near its 10⁻³ mbar ultimate; the diffusion and turbo pumps only start working where the rotary gives up. The team covers the whole range; neither member can.

07Roughing, backing, foreline, hi-vac: the names

The same rotary pump does two different jobs at different times, and the names describe the job, not the hardware:

Term	Meaning	Why it is called that
Roughing	Pumping the chamber directly with the rotary pump from atmosphere down to ~10⁻² mbar, through the roughing valve (RV) and roughing line, bypassing the diffusion pump.	It takes the chamber through the “rough” vacuum range; the gross bulk of the air is removed.
Backing	The rotary pump holding the outlet of the diffusion (or turbo) pump below its critical pressure, through the backing valve (BV).	It “backs up” the high-vacuum pump, which cannot exhaust to atmosphere on its own. The same pump is then called the backing pump.
Foreline	The pipe between the diffusion pump outlet and the backing valve/rotary pump.	The line “before” (fore) the rotary pump. Its pressure is what GH2 usually reads.
Hi-vac (high vacuum)	The diffusion or turbo pump, and the large hi-vac valve (HV) — a butterfly, gate or baffle valve — between it and the chamber.	It takes the chamber into the high-vacuum range (10⁻³–10⁻⁷ mbar).
Air admittance / vent valve	Lets air or dry nitrogen into the chamber to bring it back to atmosphere.	It “admits” air.
Crossover pressure	The chamber pressure at which you switch from roughing to the hi-vac pump (~5×10⁻² mbar for a diffusion pump).	Where the job “crosses over” from one pump to the other.

08Measuring vacuum: Pirani (GH1, GH2) and Penning

Below about 1 mbar there is too little force on a membrane to measure pressure mechanically. So vacuum gauges measure something that depends on the number of molecules: how well the gas carries heat away (Pirani), or how much current you get by ionising it (Penning).

The Pirani gauge

A fine tungsten or platinum wire is kept hot (typically ~120 °C) inside a small tube connected to the vacuum. It loses heat in three ways: radiation, conduction through its supports, and conduction through the gas. Only the last depends on pressure. The gauge's electronics keep the wire at constant temperature in a Wheatstone bridge, and the power this needs is converted into a pressure reading.

The physics of the response is a lovely piece of kinetic theory. When the mean free path is much longer than the gap between wire and wall (low pressure), each molecule that hits the wire carries away a fixed amount of energy, so the heat loss is proportional to the impingement rate, i.e. to $p$. When the mean free path is much shorter than the gap (high pressure), the thermal conductivity of a gas,

$$\kappa \approx \tfrac13\, n\, m\, c_v\, \bar v\, \lambda, \qquad \lambda \propto \frac1n \;\Rightarrow\; \kappa \text{ independent of pressure,}$$

Here $\kappa$ is the thermal conductivity of the gas, $n$ its number density, $m$ the molecular mass, $c_v$ the specific heat per unit mass at constant volume, $\bar v$ the mean speed and $\lambda$ the mean free path. Because $\lambda$ falls as $1/n$, the product $n\lambda$ — and so $\kappa$ — does not depend on pressure.

— Maxwell's surprising result — so the gauge saturates. Between the radiation floor at about $10^{-3}$ mbar and saturation at some tens of mbar, the Pirani gauge reads well. Because different gases carry heat differently, a Pirani gauge calibrated for nitrogen reads wrong in argon, helium or water vapour.

Figure 9 · Interactive

Pirani response (a simple model with typical gas factors). The useful range is where the curves slope. In argon at a true 0.1 mbar an N₂-calibrated gauge reads low, so you would think you had pumped further than you have. Always check the manufacturer's gas-correction table when you work with process gases.

GH1 and GH2

Many coating units, including those common in Indian labs, have a two-channel digital Pirani gauge whose two gauge heads are labelled GH1 and GH2. One head sits on the chamber side (on the chamber or on the roughing line next to it) and reads the chamber pressure during roughing; the other sits on the foreline and tells you whether the diffusion pump is properly backed. The usual assignment is GH1 → chamber/roughing and GH2 → foreline/backing, but manufacturers differ: follow the cable and the tube from each head on your own unit and write the answer on the panel. You need both readings — GH1 to know when to cross over, GH2 to know the diffusion pump is safe.

The Penning (cold-cathode) gauge

Below $10^{-3}$ mbar the Pirani is blind, and we switch to an ionisation gauge. A Penning gauge has a ring or rod anode at +2 to +3 kV between two cathode plates, all inside a permanent magnet. A stray electron is accelerated by the electric field but bent into a tight helix by the magnetic field, so it travels metres before it reaches the anode. On the way it ionises gas molecules; the ions are collected at the cathode. The discharge current follows

$$I = K\,p^{\,n}, \qquad n \approx 1.1\text{–}1.2$$

Here $I$ is the discharge current, $p$ the pressure, $K$ a gauge- and gas-dependent constant and $n$ an empirical exponent.

with $K$ found by calibration. Range: about $10^{-2}$ to $10^{-7}$ mbar (inverted-magnetron designs reach $10^{-11}$). Switch it on only below $10^{-2}$ mbar: at higher pressure the discharge becomes an intense glow that sputters the electrodes and contaminates the gauge. At very low pressure it can take seconds to minutes to “strike”. Hot-cathode (Bayard–Alpert) gauges use a heated filament instead and reach UHV.

Figure 10

The two gauge heads on a coating unit. The Pirani measures heat conduction (rough and medium vacuum); the Penning measures an ionisation current (high vacuum).

09The sequence, and why it matters

We now have everything needed to derive the operating procedure. Three hard constraints come straight from section 6:

The diffusion pump's outlet must stay below its critical backing pressure (~0.4 mbar) whenever it is hot. Otherwise it stalls and backstreams oil.
Its inlet must never see a pressure above ~0.1 mbar while it is hot. Otherwise the jets collapse and the hot oil meets air.
The rotary pump has only one inlet. It can rough the chamber or back the diffusion pump, not both at once (if both valves are open, chamber air at high pressure flows straight into the foreline).

The procedure is simply the order of operations that never violates any of them.

Figure 11 · Interactive simulator

A thermal-evaporation coating unit. Click a valve (or use the buttons) to open or close it; switch on the water, rotary pump and diffusion-pump heater. Gas flow is calculated from the throughputs $Q = C\Delta p$ and $Q = Sp$ on three volumes (chamber 40 L, foreline 1.5 L, roughing manifold 0.5 L), including outgassing, so pressures behave as they would on a real machine — including when you do things in the wrong order. Press Show me to watch the correct start-up. Pipe colour shows pressure (red high, blue low).

Start-up (cold machine, chamber at atmosphere)

Cooling water on. The diffusion-pump walls must be cold before the oil boils, or the vapour is not condensed and the pump overheats.
Rotary pump on, all valves closed. Let it run a minute (open the gas ballast if it has been idle or the oil may be wet).
Open the backing valve (BV). The rotary evacuates the diffusion pump and the foreline. Wait until GH2 reads below ~0.1 mbar.
Diffusion-pump heater on. Wait the warm-up time (15–30 min). Meanwhile load your substrates and source — the chamber is still isolated from the pumps by the closed hi-vac valve.
Close BV, open the roughing valve (RV). The rotary now pumps the chamber directly. The diffusion pump is briefly unbacked; that is fine for a few minutes because, with its hi-vac valve shut, almost no gas enters it, and the foreline pressure rises only slowly (watch GH2).
When GH1 reads below ~0.05 mbar (the crossover pressure): close RV, open BV. The diffusion pump is backed again.
Open the hi-vac valve (HV) slowly. The chamber pressure drops by decades within seconds; GH2 rises briefly as the diffusion pump pushes the remaining chamber gas into the foreline.
Switch on the Penning gauge once below $10^{-2}$ mbar, and wait for base pressure (for evaporation, typically $2\times10^{-5}$–$5\times10^{-6}$ mbar). A cold trap filled with liquid nitrogen, if fitted, speeds up the last part by freezing out water vapour.

Shut-down

Close the hi-vac valve. The diffusion pump is now isolated and stays backed.
Wait for the sources and substrates to cool (≥ 10–15 min). Hot metal and hot boats oxidise when air comes in.
Vent the chamber through the air-admittance valve, preferably with dry nitrogen (section 10). Close the vent valve once at atmosphere and unload.
If the machine will be used again soon, close the chamber and rough it (keeping it under vacuum keeps it clean and dry). Otherwise: diffusion-pump heater off, keep BV open and the water running until the pump is cool (~30–45 min), then close BV, stop the rotary (vent its inlet if it has no automatic valve), and turn off the water.

What goes wrong, and why

Mistake	What happens physically
Opening HV with the chamber at atmosphere (or above ~0.1 mbar) and the DP hot	A blast of air enters the pump: the jets collapse, hot oil oxidises into varnish, oil vapour is blown up into the chamber and coats everything. The most expensive mistake on the list.
Roughing for too long with BV closed and the DP hot	The foreline creeps above the critical backing pressure; the pump stalls and backstreams. Watch GH2 while roughing, and alternate RV/BV if roughing is slow.
RV and BV open together at high chamber pressure	Chamber air rushes into the foreline and the DP stalls.
Heater on without cooling water	Oil vapour is not condensed on the walls; the pump overheats until its thermal cut-out trips, and oil migrates into the foreline and the chamber.
Venting with HV open	Nitrogen or air floods the hot DP — the same failure as the first row, from the other side.
Stopping the rotary with a valve open	The pressure difference pushes rotary oil back up the line (“suck-back”) into the foreline or chamber.
Penning on above 10⁻² mbar	An intense glow discharge sputters and contaminates the gauge; the reading is meaningless.

On a turbo-pumped system the rules are gentler: the turbo can usually be started together with the rotary from atmosphere (it spins up as the pressure falls), and there is no hot oil to protect. You still must not vent it at full speed (the sudden gas load can crash the rotor), and it still needs its backing pump running.

10Venting and purging

Venting is bringing a chamber back to atmospheric pressure. Purging means replacing an unwanted gas in a space with a clean, inert one. They overlap in practice:

Venting with dry nitrogen instead of room air. Water molecules stick to metal surfaces in many layers and desorb slowly (they are what limits your base pressure, section 5). Nitrogen barely adsorbs and leaves quickly. A chamber vented with dry N₂ and opened only briefly pumps down to the same base pressure noticeably faster than one vented with humid monsoon air. Keeping a slight N₂ flow while the chamber is open (a “purge”) reduces water uptake further.
Pump–purge cycles for gas lines and process chambers (sputtering, CVD, glove boxes): pump down to $p_\text{low}$, back-fill with pure gas to $p_\text{high}$, repeat. Each cycle dilutes the impurity by $p_\text{low}/p_\text{high}$, so after $N$ cycles $$x_N = x_0\left(\frac{p_\text{low}}{p_\text{high}}\right)^{N}$$
Here $x_0$ is the initial fraction of the impurity (e.g. air) in the gas, $x_N$ the fraction after $N$ pump–purge cycles, and $p_\text{low}$, $p_\text{high}$ the pumped and back-filled pressures.
Three cycles between 1 mbar and 1000 mbar reduce air from 100 % to one part in $10^{9}$. Flow-purging alone (sweeping gas through) is far less efficient, because it only dilutes exponentially with the number of volume exchanges, and dead legs never get swept.
Gas-ballast purging of the rotary pump (section 6.1) to strip condensed vapour from the oil.
Purging the diffusion pump after maintenance or a bad venting: never with hot oil; vent the cold pump with dry N₂.

11Thermal evaporation

Thermal (resistive) evaporation is the simplest PVD method: put the material in a boat, filament or crucible made of a refractory metal (W, Mo, Ta), pass a large current through it, and let the material melt and evaporate. The vapour travels in straight lines (we made sure of that with the vacuum) and condenses on everything in sight, including the substrate.

Figure 12

Inside a thermal evaporator. The boat between the copper LT electrodes is the source. The source shutter lets you melt and outgas the charge without coating anything; the substrate shutter lets the crystal monitor see the rate stabilise before the film starts. The HT ring is used for glow-discharge cleaning before deposition (section 12).

How much heat: Joule heating with the LT supply

A boat is a resistor of a few to a few tens of milliohms. The heating power is

$$P = I^2 R$$

Here $P$ is the electrical power dissipated in the boat, $I$ the current through it and $R$ its resistance. In the radiation term below, $\varepsilon$ is the emissivity of the hot surface, $\sigma = 5.67\times10^{-8}$ W m⁻² K⁻⁴ the Stefan–Boltzmann constant, $A$ the radiating area and $T$ the absolute temperature.

so to deliver several hundred watts into 15 mΩ you need ~150–250 A at only 2–5 V. That is what the step-down transformer of the LT (low-tension) supply provides (section 12). The boat reaches a temperature where the input power balances radiation ($\varepsilon\sigma A T^4$) and conduction into the water-cooled electrodes. Because $R$ itself rises with temperature, you raise the current slowly — both to avoid thermal shock and to let the charge outgas.

Figure 13 · Interactive

The LT circuit in numbers. A few volts and a few hundred amps on the boat side; a few amps from the 230 V mains on the primary side.

How fast a metal evaporates: vapour pressure and Hertz–Knudsen

Every solid and liquid has a vapour pressure $p_v(T)$: the pressure of its own vapour in equilibrium with it. Thermodynamics (the Clausius–Clapeyron equation) gives its temperature dependence:

$$\frac{d\ln p_v}{dT} = \frac{\Delta H_\text{vap}}{RT^2} \quad\Rightarrow\quad \log_{10} p_v = A - \frac{B}{T}$$

Here $p_v$ is the vapour pressure, $T$ the absolute temperature, $\Delta H_\text{vap}$ the enthalpy of vaporisation (or of sublimation, for a solid) per mole, $R = 8.314$ J mol⁻¹ K⁻¹ the gas constant, and $A$, $B$ constants tabulated for each material ($B = \Delta H_\text{vap}/(2.303\,R)$).

— an exponential rise with temperature. In equilibrium, the flux of atoms leaving the surface equals the flux of vapour atoms arriving (section 2, equation 3). In vacuum nothing comes back, so the surface keeps emitting at the equilibrium rate. That is the Hertz–Knudsen–Langmuir equation:

$$\Gamma = \frac{\alpha\,(p_v - p)}{\sqrt{2\pi m k_B T}} \qquad \Gamma_m = m\Gamma = \alpha\,p_v\sqrt{\frac{M}{2\pi R T}}$$

Γ: atoms per m² per s · Γ_m: kg m⁻² s⁻¹ · α ≤ 1 evaporation coefficient · M molar mass

Here $p_v$ is the vapour pressure of the source at its temperature $T$, $p$ the partial pressure of the same vapour above the surface (≈ 0 in high vacuum), $m$ the mass of one evaporating atom or molecule, $M$ its molar mass, $R$ the gas constant and $k_B$ Boltzmann's constant.

In practical units, $\Gamma_m \approx 5.83\times10^{-2}\, p_v[\mathrm{Torr}]\sqrt{M/T}$ g cm⁻² s⁻¹. A useful rate needs $p_v \approx 10^{-2}$ mbar, which sets the evaporation temperature of each material. Some metals (Cr, Ti at modest temperatures) reach it below their melting point and sublime from the solid.

Figure 14 · Interactive

Vapour pressures of common metals (fitted to standard vapour-pressure tables; dots mark the melting points). The calculator applies Hertz–Knudsen to a 1 cm² melt and the cosine law below to estimate the film growth rate straight above it. Notice how steep the curves are: 100 °C more can mean ten times the rate.

Where the atoms land: the cosine law and film uniformity

A small flat source emits more atoms straight up than sideways. Knudsen showed that the number leaving at angle $\theta$ from the surface normal goes as $\cos\theta$ (a point source would emit equally in all directions). An area element of the substrate at distance $r$, whose normal makes an angle $\phi$ with the incoming direction, receives

$$\frac{dM_s}{dA_s} = \frac{M_e}{\pi r^2}\cos\theta\,\cos\phi$$

Here $M_e$ is the total mass evaporated from the source, $dM_s$ the mass landing on a small substrate area $dA_s$, $r$ the source–substrate distance, $\theta$ the emission angle measured from the source normal and $\phi$ the angle of arrival measured from the substrate normal.

For a flat substrate parallel to the source at height $h$, a point at lateral distance $\ell$ has $\cos\theta = \cos\phi = h/r$ and $r^2 = h^2 + \ell^2$. Dividing by density gives the thickness, and normalising to the thickness $t_0$ directly above the source:

$$\frac{t}{t_0} = \frac{1}{\big[1 + (\ell/h)^2\big]^{2}}\ \ \text{(small surface source)}, \qquad \frac{t}{t_0} = \frac{1}{\big[1 + (\ell/h)^2\big]^{3/2}}\ \ \text{(point source)}$$

film uniformity on a flat substrate

Here $t$ is the film thickness at lateral distance $\ell$ from the point directly above the source, $t_0$ the thickness at that point, and $h$ the source–substrate height.

With $h = 25$ cm, a point 5 cm off-centre gets 92 % of the centre thickness. To do better you increase $h$ (at the cost of rate, which falls as $1/h^2$), rotate the substrates, or put them on a sphere passing through the source (on such a Knudsen sphere, $\cos\theta\cos\phi/r^2$ is constant and the coating is perfectly uniform for a surface source).

Figure 15 · Interactive Monte Carlo

Building the thickness profile atom by atom. Each frame launches thousands of atoms with the chosen angular distribution and records where they land on a flat substrate; the bars converge onto the analytical curve (cyan). This geometry is also why evaporation is directional — the property that makes lift-off work (section 15).

Practical notes

Boat choice: W for most metals, Mo or Ta for others; alumina-coated boats or crucibles for metals that alloy with or wet the boat (Al dissolves tungsten — use coated boats or BN-TiB₂ crucibles). Au, Ag, Cu, Cr, Al are routine; refractory metals (W, Mo, Pt, Ta) and most oxides are not possible thermally — that is what e-beam is for.
Alloys do not evaporate at their composition: the component with the higher vapour pressure leaves first (Raoult's law), so the film composition drifts.
Adhesion layers: noble metals stick poorly to glass and oxides. A 2–5 nm Cr or Ti layer (both oxide-forming) is deposited first.
Contamination from the boat is the main purity limit, along with the residual gas.

12LT current and HT voltage

“Tension” is the old British (and Indian) engineering word for voltage. LT is low tension; HT is high tension. On a coating unit they are two very different supplies for two very different jobs:

	LT (low tension)	HT (high tension)
Voltage	a few volts (typically 0–10 V)	kilovolts (typically 0–3 kV on a thermal unit; 5–10 kV on an e-beam gun)
Current	tens to hundreds of amperes	milliamperes (thermal unit) to ~1 A (e-beam)
What it drives	the resistive boat or filament (thermal evaporation); the e-beam gun's filament	glow-discharge cleaning ring (thermal unit); the electron acceleration in an e-beam gun
Physics	Joule heating $P = I^2R$ in a low resistance	accelerating charges: $E = eV$; striking a gas discharge
Built from	step-down transformer (and variac or thyristor control)	step-up transformer and rectifier
Hazard	very hot electrodes and boats; huge currents through loose contacts	lethal voltage; interlocks must never be bypassed

Same power, opposite proportions: 3 V × 200 A = 600 W from LT; 3 kV × 0.2 A = 600 W from HT. The LT current heats; the HT voltage accelerates.

HT on a thermal unit: glow-discharge cleaning and the Paschen law

Before deposition, at a pressure of about $10^{-1}$–$10^{-2}$ mbar (during roughing, or with a little gas admitted through a needle valve), the HT supply puts 1–3 kV on an aluminium ring electrode. A glow discharge strikes: gas ions accelerate into the surfaces and knock off adsorbed water and organic contamination, improving film adhesion dramatically.

Why only in that pressure window? A discharge needs each electron to ionise enough gas molecules on its way across the gap $d$ to replace itself. The ionisation per unit length is Townsend's $\alpha = Ap\,e^{-Bp/E}$ and the breakdown condition $\gamma(e^{\alpha d}-1) = 1$ ($\gamma$: secondary electrons per ion at the cathode) leads to Paschen's law:

$$V_b = \frac{B\,pd}{\ln(A\,pd) - \ln\!\big[\ln(1 + 1/\gamma)\big]}$$

breakdown voltage depends only on the product p·d

Here $V_b$ is the breakdown voltage, $p$ the gas pressure, $d$ the electrode gap, $E$ the electric field, $A$ and $B$ constants of the gas (from Townsend's ionisation coefficient $\alpha$, the number of ionising collisions an electron makes per unit length), and $\gamma$ the number of secondary electrons released per ion striking the cathode.

At high pressure, electrons collide too often to gain ionising energy, so a huge voltage is needed. At low pressure, they cross the gap without meeting a molecule at all, and no voltage helps. In between lies a minimum (a few hundred volts for air at $pd \approx 1$ Torr·cm). With a gap of ~10 cm, that minimum sits near $10^{-1}$ mbar — exactly the glow-cleaning window. It is also why a Penning gauge can run a discharge at $10^{-6}$ mbar only because its magnet makes the electrons' path, not the gap, long.

Figure 16 · Interactive

Paschen curve for air (A = 15 cm⁻¹Torr⁻¹, B = 365 V cm⁻¹Torr⁻¹, γ = 0.01). Move the pressure down from atmosphere: the glow strikes only inside the window where the curve dips below the HT supply voltage, and goes dark again in high vacuum.

13Electron-beam evaporation

Thermal evaporation heats the whole boat, so the boat must be hotter than the material — which limits you to materials that evaporate below ~1800 °C and do not react with the boat. Electron-beam evaporation turns the problem around: heat only the surface of the material itself, with a focused beam of energetic electrons, while it sits in a water-cooled copper hearth that stays cold.

Step 1 — making electrons: thermionic emission (LT)

A tungsten filament is heated by an LT current to ~2500 K. Some of its electrons then have enough thermal energy to escape over the work function $W$ (4.5 eV for W). The emitted current density follows the Richardson–Dushman equation:

$$J = A_G\,T^2\,e^{-W/k_B T}, \qquad A_G \approx 0.5\text{–}1.2\times10^{6}\ \mathrm{A\,m^{-2}\,K^{-2}}$$

Here $J$ is the emitted current density, $T$ the filament temperature, $W$ the work function (the energy needed to remove an electron from the metal), $k_B$ Boltzmann's constant and $A_G$ the material-dependent Richardson constant.

The exponential makes emission extremely sensitive to filament temperature, which is how the emission current (and so the beam power) is controlled.

Step 2 — accelerating them (HT)

The electrons are accelerated through a potential of $V$ = 5–10 kV, gaining kinetic energy $E = eV$. At 10 kV their speed is about 0.2 c, so a small relativistic correction already appears:

$$\gamma = 1 + \frac{eV}{m_e c^2}, \qquad p = \frac{1}{c}\sqrt{(eV)^2 + 2\,eV\,m_ec^2}$$

Here $\gamma$ is the relativistic Lorentz factor (not the secondary-electron yield of the previous section), $p$ the electron's momentum (not pressure), $e$ the elementary charge, $V$ the accelerating voltage, $m_e$ the electron rest mass and $c$ the speed of light; $m_ec^2 = 511$ keV.

Step 3 — bending them 270°

The filament is hidden below and beside the hearth so that evaporated material cannot coat it and it does not see the hot melt. A permanent magnet (plus coils) supplies a field $B$ perpendicular to the beam. The Lorentz force $\mathbf F = -e\,\mathbf v\times\mathbf B$ ($\mathbf v$: electron velocity, $\mathbf B$: magnetic field, × the vector cross product) bends the electrons into a circle of radius

$$r = \frac{p}{eB} \;\approx\; \frac{\sqrt{2 m_e eV}}{eB}$$

Here $r$ is the radius of the electron's circular path, $p$ its momentum, $e$ its charge and $B$ the magnetic flux density; the right-hand form neglects the small relativistic correction.

With 8 kV and 7.6 mT, $r \approx 4$ cm: the beam loops up and over through 270° and lands in the crucible from above. Extra coils sweep the spot across the melt so it heats evenly and does not drill a hole.

Figure 17 · Interactive

A 270° electron gun. The trajectory is calculated from $r = p/(eB)$ with the relativistic momentum. Raise the voltage without raising the field and the beam lands beyond the melt — which is why the high voltage and the magnet are set together, and why the beam position is corrected with the deflection coils on a real gun.

Step 4 — what happens at the melt

The beam delivers power $P = V\cdot I$ (accelerating voltage times beam current): 8 kV × 0.3 A = 2.4 kW into a spot a few millimetres across, a power density of order $10^4$ W cm⁻². The surface melts and evaporates; the copper hearth, cooled by water, keeps the outer part of the charge solid, so the melt sits in a “skull” of its own material and never touches anything else. That is why e-beam films are purer, and why it can evaporate W, Mo, Ta, Pt, and oxides such as SiO₂, TiO₂, Al₂O₃ and HfO₂. Crucible liners (graphite, Mo, alumina) are used for metals that would otherwise wet the copper or conduct too much heat away.

X-rays

Stopping 10 keV electrons in metal produces bremsstrahlung X-rays up to 10 keV. The chamber walls absorb them, but windows must be leaded glass, and X-rays plus stray electrons can damage sensitive devices (oxides in transistors, organic layers) on the substrate. This is one reason thermal evaporation is still preferred for some organic and device work.

	Thermal (resistive)	Electron-beam
What is heated	the boat, and the charge by contact	only the charge surface
Supply	LT: 2–5 V, 100–300 A	LT filament + HT 5–10 kV, 0.05–1 A
Max. practical temperature	~1800 °C (boat limit)	> 3000 °C at the spot
Materials	Au, Ag, Al, Cu, Cr, In, organics, some fluorides	almost anything: W, Mo, Ta, Pt, Ti, Ni, oxides
Purity	boat contamination possible	high (cold hearth)
Rate control	via LT current, slow response	via emission current, fast response
Side effects	radiant heat on substrates	X-rays, stray electrons, charging
Cost and complexity	low	high

14The DTM: weighing a film with a quartz crystal

How do you know you have deposited 50 nm and not 45? The digital thickness monitor (DTM), also called a quartz crystal monitor (QCM) or film thickness monitor, weighs the film as it grows, with a resolution far below one atomic layer. It does this with a vibrating crystal.

The crystal

Quartz is piezoelectric: a voltage deforms it, and deforming it produces a voltage. A thin disc cut at a particular angle to the crystal axes (the AT-cut) vibrates in thickness-shear mode — its two faces slide back and forth relative to each other — when an alternating voltage is applied to gold electrodes on both faces. The fundamental resonance is the frequency at which the plate is half an acoustic wavelength thick:

$$f_0 = \frac{v_s}{2\,t_q} = \frac{N_q}{t_q}, \qquad N_q = 1.668\times10^{5}\ \mathrm{Hz\,cm}$$

Here $f_0$ is the fundamental resonance frequency, $v_s$ the speed of the shear acoustic wave in quartz, $t_q$ the thickness of the quartz plate and $N_q = v_s/2$ the frequency constant of AT-cut quartz.

A 6 MHz crystal is therefore 0.278 mm thick. The AT-cut is chosen because its frequency barely changes with temperature near room temperature; the sensor head is still water-cooled, because radiant heat from the source would otherwise drift the reading.

Figure 18

The DTM chain. The film adds mass to the vibrating plate, its resonance frequency falls, the oscillator follows it, and a frequency counter converts the shift into a thickness using the film's density.

The Sauerbrey equation

A thin, rigid film of mass $\Delta m$ on an area $A$ behaves as extra quartz: it makes the plate effectively thicker, so the frequency falls. Differentiating $f_0 = N_q/t_q$ and writing the added thickness as mass gives the Sauerbrey equation (1959):

$$\Delta f = -\frac{2 f_0^2}{\sqrt{\rho_q\,\mu_q}}\,\frac{\Delta m}{A} = -\frac{f_0^2}{N_q\,\rho_q}\,\frac{\Delta m}{A}$$

ρ_q = 2.648 g cm⁻³, μ_q = 2.947×10¹¹ g cm⁻¹ s⁻² (shear modulus of quartz)

Here $\Delta f$ is the change in resonance frequency, $f_0$ the resonance frequency before deposition, $\Delta m$ the deposited mass, $A$ the active (electrode) area of the crystal, $\rho_q$ the density and $\mu_q$ the shear modulus of quartz.

For a 6 MHz crystal the sensitivity is 81.5 Hz per µg cm⁻². One ångström of gold ($19.3\ \mathrm{g\,cm^{-3}}\times10^{-8}$ cm $=1.93\times10^{-7}$ g cm⁻²) lowers the frequency by about 16 Hz; one ångström of aluminium by 2.2 Hz. A counter resolving 0.1 Hz therefore sees a small fraction of an atomic layer. The thickness follows from the mass per area:

$$t_f = \frac{\Delta m}{\rho_f\,A}$$

Here $t_f$ is the film thickness and $\rho_f$ the film density.

— which is why you must enter the density of the material in the DTM. Enter the density of gold while depositing aluminium and the reading is wrong by a factor of seven.

The Z-ratio: when the film is no longer “thin”

Sauerbrey assumes the film is acoustically identical to quartz. As the film thickens, the acoustic wave travels partly in a material with a different acoustic impedance $Z = \sqrt{\rho\mu}$, and the frequency–mass relation bends. Lu and Lewis (1972) solved the composite resonator exactly:

$$t_f = \frac{N_q\,\rho_q}{\pi\,\rho_f\,f_c\,Z}\arctan\!\Big[Z\tan\Big(\pi\,\frac{f_q - f_c}{f_q}\Big)\Big], \qquad Z = \sqrt{\frac{\rho_q\mu_q}{\rho_f\mu_f}}$$

f_q: uncoated crystal frequency · f_c: coated · Z: the “Z-ratio” you enter

Here $f_q$ is the frequency of the uncoated crystal, $f_c$ the frequency of the coated crystal, $\rho_f$ and $\mu_f$ the density and shear modulus of the film, and $Z$ the ratio of the acoustic impedances of quartz and film.

For $Z = 1$ it reduces to the Sauerbrey (period) form. Gold has $Z = 0.381$, so on a crystal that has collected a lot of gold the simple formula overestimates the thickness; modern monitors use the Z-corrected formula, which is why they ask for the Z-ratio as well as the density.

The tooling factor

The crystal cannot sit where the substrate is: it is off to one side, at a different distance and angle from the source. By the cosine law it receives a different flux. The tooling factor corrects for this geometry:

$$\text{TF}\,(\%) = 100\times\frac{t_\text{substrate}}{t_\text{crystal}}, \qquad \text{TF}_\text{new} = \text{TF}_\text{old}\times\frac{t_\text{measured}}{t_\text{DTM}}$$

Here $t_\text{substrate}$ and $t_\text{crystal}$ are the thicknesses deposited at the same time on the substrate and on the crystal, $t_\text{measured}$ the real film thickness measured afterwards, and $t_\text{DTM}$ the thickness the monitor displayed.

Calibrate it once per source position and material: deposit a film, measure its real thickness with a profilometer, AFM or ellipsometer, and update the tooling factor with the second formula.

Figure 19 · Interactive DTM

Run a deposition on a simulated DTM. The crystal's frequency is computed from the film on it with the Lu–Lewis equation. The true geometry is the real substrate-to-crystal flux ratio; the tooling factor entered is what the monitor believes. Start the source with the shutter closed (as you should, to let the rate settle), zero, open the shutter, and compare the display with what actually reached the substrate. Then set the entered tooling factor equal to the true one and repeat.

Using the DTM well

Enter density, Z-ratio and tooling for the material before you start. Thin-film densities can be a few percent below bulk.
Bring the source up with the shutter closed; watch the rate on the crystal, which sees the source; open the substrate shutter only once the rate is steady, and zero the thickness at that moment. Close the shutter at the target thickness, then ramp the power down.
Many monitors close the loop: a PID controller adjusts the LT current or e-beam emission to hold the rate.
Crystal life: as mass builds up, the resonance weakens and eventually fails (often after the frequency has fallen by roughly 1 MHz on a 6 MHz crystal, depending on the material and on film stress). Change it before an important run; a crystal that “mode-hops” gives sudden jumps in thickness.

15Electron-beam lithography (EBL)

The names are almost identical, but electron-beam lithography and electron-beam evaporation do completely different things. E-beam evaporation uses a broad, powerful beam as a heater. EBL uses a finely focused, very weak beam as a pen: it draws a pattern into a thin polymer film called a resist, changing the polymer's chemistry where the beam has been. No material is deposited by EBL itself.

The column

An EBL system is essentially a scanning electron microscope with a fast beam blanker, a pattern generator and a precision stage. A (usually Schottky field-emission) gun produces electrons; they are accelerated through 10–100 kV; magnetic lenses demagnify the source to a spot of a few nanometres; a blanker switches the beam on and off in nanoseconds; deflection coils scan it over a writing field of up to a millimetre; a laser-interferometer stage moves the sample between fields and stitches them together. The whole column needs $10^{-6}$–$10^{-9}$ mbar: electrons must travel more than a metre without scattering, and the field-emission tip needs ultra-high vacuum to stay clean.

Figure 20

An EBL column (left) and what the electrons do in the sample (right). Electrons spread a little in the resist (forward scattering) and a lot in the substrate, from which some are scattered back up into the resist micrometres away. That second effect is the proximity effect.

Why not light? Wavelength is not the limit

Optical lithography is limited by diffraction to roughly the wavelength of light. Electrons have a de Broglie wavelength

$$\lambda_e = \frac{h}{\sqrt{2 m_e eV\left(1 + \dfrac{eV}{2m_ec^2}\right)}}$$

Here $\lambda_e$ is the electron wavelength, $h$ Planck's constant, $m_e$ the electron rest mass, $e$ its charge, $V$ the accelerating voltage and $c$ the speed of light; the bracket is the relativistic correction.

which is picometres at EBL voltages — tens of thousands of times shorter than UV light. The resolution of EBL (~5–20 nm) is set not by diffraction but by lens aberrations, electron scattering in the resist, the secondary electrons that actually break the polymer bonds, and the resist development.

Figure 21 · Interactive

Electron wavelength. At 100 kV the relativistic correction is already 5 %.

Resists, dose and development

The workhorse resist is PMMA (poly(methyl methacrylate), the polymer of acrylic glass), spin-coated to 50–500 nm. Electrons break its long chains (chain scission); the shorter chains dissolve in a developer (MIBK:IPA 1:3) while the unexposed long chains do not. That makes PMMA a positive resist: exposed areas are removed. Negative resists such as HSQ cross-link instead, and exposed areas remain.

The exposure is measured as dose $D$, charge per unit area. The minimum dose at which the resist clears completely is the clearing dose, about 100–400 µC cm⁻² for PMMA at 20–30 kV and higher at 100 kV (faster electrons deposit less energy per nanometre in the thin resist). With a beam current $I$, the exposure time for an area $A$ is simply

$$t = \frac{D\,A}{I}$$

Here $t$ is the exposure time, $D$ the dose (charge per unit area), $A$ the exposed area and $I$ the beam current.

and each pixel of size $s$ gets a dwell time $t_d = D s^2/I$. EBL is a serial technique, which is its great weakness: one square millimetre at 300 µC cm⁻² with 1 nA takes 50 minutes of pure exposure.

Figure 22 · Interactive

Write time and dwell time. High currents write faster but give a larger spot; small pixels need a pattern generator that can blank and step at many megahertz.

The proximity effect

The energy deposited in the resist by a single point exposure is well described by a sum of two Gaussians (Chang, 1975): a narrow one for forward scattering and a wide, weaker one for back-scattered electrons:

$$f(r) = \frac{1}{\pi(1+\eta)}\left[\frac{1}{\alpha^2}e^{-r^2/\alpha^2} + \frac{\eta}{\beta^2}e^{-r^2/\beta^2}\right]$$

α: forward-scatter range (nm) · β: back-scatter range (µm) · η: back-scattered energy ÷ forward

Here $f(r)$ is the energy deposited per unit area at distance $r$ from the point where the beam enters the resist, normalised so that its integral over the plane is 1; $\alpha$, $\beta$ and $\eta$ are given below the equation.

Because $\eta$ is of order 0.5–0.8 on silicon and $\beta$ is micrometres, every exposed line also delivers a background dose to its neighbours. In the middle of a dense grating the background adds up; at the edge it does not. The same dose then gives wider lines in the centre than at the edge, and isolated features need a higher dose than dense ones. Proximity-effect correction adjusts the dose of every shape so that the absorbed dose is the same everywhere.

Figure 23 · Interactive

Proximity effect on a line grating. The absorbed dose (gold) is the designed pattern (grey) convolved with the double-Gaussian point-spread function. Where it exceeds the clearing dose (red), the resist develops away (green). Increase the number of lines or the back-scatter ratio and the centre lines grow wider than the edge lines.

Lift-off: where lithography meets evaporation

To make metal nanostructures (contacts, plasmonic antennas, gratings), the EBL pattern is transferred by lift-off: deposit metal over the developed resist, then dissolve the resist so the metal on top of it floats away, leaving metal only where the resist was opened. For this to work, the metal film on the resist and the film in the openings must not be connected. That needs a directional deposition from a small, distant source — thermal or e-beam evaporation, with its cosine-law beam in high vacuum (sections 3 and 11) — and an undercut resist profile, usually made with a two-layer PMMA stack whose bottom layer is more sensitive. Sputtering, at $10^{-3}$ mbar with a short mean free path and a large target, coats the side walls and ruins lift-off.

Figure 24 · Interactive

EBL and lift-off, step by step. Switch the deposition method to sputtering at step 4 to see why lift-off processes use evaporation.

16Thermal vs e-beam evaporation vs EBL

	Thermal evaporation	E-beam evaporation	E-beam lithography
Purpose	deposit a film	deposit a film	draw a pattern in resist
Energy source	Joule heating (LT current)	electron beam as a heater	electron beam as a pen
Electron energy	—	5–10 keV	10–100 keV
Beam current	—	0.05–1 A	pA – tens of nA
Power	0.2–2 kW in the boat	1–10 kW in the melt	~0.1 mW in the sample
Spot size	whole boat (cm)	mm	nm
Working pressure	10⁻⁵–10⁻⁶ mbar	10⁻⁶–10⁻⁷ mbar	10⁻⁶–10⁻⁹ mbar (column)
Result	uniform film over the substrate	uniform film over the substrate	chemically changed resist; no material added
Typical next step	—	—	develop → evaporate → lift-off, or etch

The power column is the clearest way to see the difference: an e-beam evaporator puts ten million times more power into its target than an EBL tool does.

17A complete run: 5 nm Cr + 50 nm Au

Here is a full thermal-evaporation run, with the reason for each step, putting the whole article together. (Generic settings; use your own instrument's manual and your lab's rules.)

Prepare substrates: solvent clean (acetone, IPA), dry with N₂, optionally oxygen plasma. Every monolayer of contamination you bring in competes with your film (section 3).
Load: Cr (chips or a Cr-coated rod) in one boat, Au wire in another (a W boat for each). Check boat continuity with the LT off. Mount substrates face-down, note the distance to the sources, check the shutters move freely.
Pump down following section 9. Glow-clean with HT at ~0.1 mbar for a few minutes during roughing if your unit has a glow ring (section 12).
Wait for base pressure, ideally ≤ 5×10⁻⁶ mbar. At that pressure the mean free path is over 10 m and the monolayer time about a second (section 3).
Set the DTM for chromium: density 7.20 g cm⁻³, Z 0.305, tooling factor from your last calibration.
Chromium: shutter closed, raise the LT current slowly; Cr sublimes before it melts (section 11). When the rate is steady at ~0.5 Å/s, zero the DTM, open the shutter; close it at 5 nm. Ramp the current down.
Gold: switch the DTM to Au (19.32 g cm⁻³, Z 0.381). Switch the LT output to the Au boat. Raise the current until the gold melts and the rate is steady at 1–2 Å/s. Zero, open, close at 50 nm. Ramp down. Depositing gold immediately after chromium, without breaking vacuum, keeps the Cr surface unoxidised so the two layers bond.
Cool ≥ 15 minutes under vacuum, close the hi-vac valve, vent with dry N₂ (section 10), unload, and pump the chamber back down or shut down (section 9).
Measure the real thickness (profilometer step, AFM, ellipsometry or X-ray reflectivity) and update the tooling factors (section 14).

18Safety

High voltage: HT supplies and e-beam guns operate at lethal kilovolts. Never open covers or bypass interlocks; ground the electrode with a grounding stick before touching it.
High current and heat: LT bus bars and boats are hot long after the current stops. Use gloves; loose connections at hundreds of amperes overheat and arc.
Implosion: a glass bell jar under vacuum stores a large force (1 bar on 0.1 m² is 10 kN). Use the implosion guard; never knock or heat it unevenly.
X-rays around e-beam guns: leaded viewports, no improvised windows.
Liquid nitrogen: cold burns, and asphyxiation in small rooms. Never seal it in a closed container.
Pump oil and exhaust: rotary exhaust carries oil mist and whatever you pumped (solvents, process gases) — vent it outdoors or through a filter.
Toxic materials: Cr, Ni, Cd, As, Se and fine metal flakes from chamber walls. Clean the chamber with gloves and a mask, not compressed air.

19Glossary and further reading

Term	Meaning
RV / BV / HV / AV	roughing, backing, hi-vac and air-admittance (vent) valves
GH1, GH2	Pirani gauge heads; typically chamber/roughing and foreline/backing (check your unit)
Penning (PG)	cold-cathode ionisation gauge for high vacuum
DP, TMP	diffusion pump, turbomolecular pump
LT / HT	low-tension (low voltage, high current) / high-tension (high voltage) supply
DTM / QCM	digital thickness monitor / quartz crystal microbalance
Tooling factor	ratio of substrate thickness to crystal thickness, in %
Z-ratio	acoustic impedance of quartz ÷ that of the film
Critical backing pressure	highest foreline pressure a diffusion pump tolerates
Crossover pressure	chamber pressure at which roughing hands over to the hi-vac pump
Base pressure	the lowest pressure reached before deposition starts
Mean free path λ	average distance a molecule travels between collisions
Clearing dose	EBL dose at which the resist develops away completely
Lift-off	patterning a deposited film by dissolving the resist under it