The bias a waveform can choose

The bias no battery supplies found a steady negative voltage on an electrode that nothing had charged. A sheath lets electrons in thousands of times more readily than it lets them out, so an electrode driven through a capacitor walks negative until the electrons it can grab on each positive swing only just balance the ions it collects all the time. Put two electrodes in the same chamber and the one with the smaller area takes most of the voltage. That area law is how the ions that etch a silicon wafer are given their energy, and it has a cost the essay ended on: in a chamber built with a given ratio of areas, the ion energy and the plasma density both follow from the one drive voltage, so they cannot be set separately.

This essay removes the geometry entirely. Both electrodes are the same size, so a sine-wave drive has no reason to push the plasma one way rather than the other, and it does not; the bias is zero. The asymmetry is put back into the waveform instead. The result is a discharge in which the direct voltage is set by a phase — a knob with no counterpart in the geometric version and no effect on how much power goes in.

Same harmonics, different extremes #

The drive is a sum of two cosines, one at the fundamental frequency and one at twice it, with equal amplitudes $\varphi_0$ and a phase θ between them. Neither term has a direct component, and the mean of their sum over a cycle is zero at every θ. The spectrum is the same at every θ too: two lines of equal height. What changes with θ is the shape.

With the two harmonics in phase their peaks coincide once a cycle and the waveform rises to twice $\varphi_0$; the troughs never coincide, and the deepest the sum reaches is −1.125 $\varphi_0$, at the point where the fundamental’s cosine is a quarter below zero. Shift the second harmonic by 90° of the fundamental and the waveform turns upside down: it dips to −2 $\varphi_0$ and peaks at only +1.125. The average is still zero and the power still the same. The asymmetry is between the highest and the lowest point, and the drawing shows that this alone is enough to choose a bias.

This is a harmonic synthesis of the kind that turns a steady glow into pulses when many modes of a laser are locked in phase. There the phases decide whether the light arrives as a smooth stream or as sharp bursts with the same total power. Here there are only two components, and the phase decides whether the waveform’s tall excursion points up or down.

A clamp on each side #

The reason the extremes matter is the one the wall a plasma builds against itself established: a sheath is a diode. Electrons cross it towards an electrode only when the electrode comes within a few electron temperatures of the plasma’s potential — the current rising as the Boltzmann exponential of how close it comes — and otherwise the sheath holds whatever voltage the circuit puts across it while a small, steady ion current trickles through. In electronics, a diode and a capacitor in series make a clamp — the circuit that restores a signal’s direct level in a television receiver by pinning its most positive point to a reference. An electrode behind a blocking capacitor is exactly that circuit, and a symmetric discharge is two of them facing each other.

Each sheath must collapse once a cycle, briefly, to let in the electrons that neutralise the ions it has collected. The sheath at the driven electrode collapses when the applied voltage is at its maximum; the sheath at the grounded electrode collapses when the applied voltage is at its minimum. Between those instants the two sheaths share the voltage, and in the simplest model of a capacitive sheath — ions of uniform density, so that the voltage across a sheath goes as the square of the charge it holds — the requirement that each touch zero at its own moment has a unique solution. The discharge settles so that the two extremes, measured from the bias, are equal and opposite. With equal electrodes the bias is minus the mean of the maximum and the minimum:

$$\eta = -\tfrac{1}{2}\left(\tilde\phi_{\max} + \tilde\phi_{\min}\right).$$

For a sine wave that is zero. For the in-phase two-frequency waveform it is −(2 − 1.125)/2 = −0.4375 $\varphi_0$, and for the waveform at 90°, +0.4375 $\varphi_0$. The electrode has been pushed negative, or positive, by a waveform whose average is zero and whose electrodes are identical, and the only thing that decided the sign was which way up the tall peak pointed.

Why odd harmonics cannot do it #

The clamp condition says at once which waveforms can bias a symmetric chamber and which cannot, and the answer is sharper than “anything that is not a sine wave”. A bias appears only when the waveform’s highest point and its lowest point differ in size. A waveform that repeats itself upside down half a cycle later — every value at one moment matched by its negative half a period on — has extremes that are exactly equal and opposite, and so no bias however distorted it looks. A square wave, a triangle wave and a sine with any amount of third, fifth or seventh harmonic added all have that half-wave symmetry, because every odd harmonic changes sign when the fundamental does. Adding them sharpens the shape and leaves the bias at zero.

An even harmonic breaks the symmetry. Half a cycle of the fundamental is a whole cycle of the second harmonic, so the second harmonic does not flip when the fundamental does, and wherever it adds to the fundamental’s peak it subtracts from the trough. The second harmonic is therefore the lowest one that can bias a symmetric discharge, and it is what the drawings use. A generator locked to twice its own frequency is also the easiest to build, which is not a coincidence the physics arranged but a convenience it permits.

The same bookkeeping explains why the phase matters and the amplitudes alone do not. The power a periodic voltage delivers into a fixed load is the sum of the powers in its harmonics, one term per line in the spectrum, and a relative phase appears nowhere in that sum. The extremes are the opposite: they are set by where the peaks of the components line up, which is exactly what the phase controls. A quantity that depends only on the spectrum cannot see the phase; a nonlinear element that responds to extremes sees little else. A pulsating field splits into two rotating ones by the same kind of reading — a superposition taken apart differently depending on what the observer is sensitive to.

Sheaths that peak alike and average differently #

The two sheath voltages over a cycle carry a detail the bias alone hides. Both sheaths reach the same peak, half the peak-to-peak voltage of the drive, because the model makes each sheath hold the whole of that when the other has collapsed. What differs is how long each spends near its peak. The driven sheath collapses for a brief instant at the waveform’s sharp positive spike and is wide open for most of the cycle; the grounded sheath collapses during the long, shallow negative stretch and is open only near the spike. Averaged over a cycle, the driven sheath holds 0.80 $\varphi_0$ and the grounded one 0.36 $\varphi_0$.

The average is what matters for the ions. A heavy ion crossing a sheath takes many radio-frequency cycles to do it, so it responds to the time-averaged field rather than the instantaneous one and arrives with an energy close to the mean sheath voltage — the regime in which the ion energy distribution is a single narrow peak. At $\varphi_0$ = 100 V the ions striking the driven electrode arrive with about 80 electronvolts and those striking the grounded electrode with about 36. The symmetric chamber has become an asymmetric one, with no change of hardware.

The two sheaths must always add up to the applied voltage plus the bias at every instant, and the drawing checks it. That is just the statement that the plasma between them is a good conductor, screening any field inside it within a Debye length, which holds as long as the plasma’s own voltage drop is small compared with the sheaths’, the usual situation in the low-pressure discharges used for etching.

A dial rather than a ratio #

Sweep the phase and the bias moves smoothly from −0.44 $\varphi_0$ at 0° to +0.44 $\varphi_0$ at 90° and back, nearly in a straight line. A chamber that is not quite symmetric — electrodes of slightly different size, or a plasma denser near one of them — has a bias that is not zero at 45°, and its dependence on phase is shifted rather than destroyed. The general form of the clamp condition weights the two extremes by a symmetry parameter ε, the ratio of the two sheaths’ voltages at equal charge:

$$\eta = -\frac{\tilde\phi_{\max} + \varepsilon\,\tilde\phi_{\min}}{1 + \varepsilon}.$$

With a single sine wave this reduces to the area law’s result, a bias of $-\varphi_0(1 - \varepsilon)/(1 + \varepsilon)$, zero when the chamber is symmetric and approaching the full amplitude when one electrode is much smaller. With two harmonics it has a geometric part and a waveform part, and they add. That additivity is why the effect was taken up quickly after Heil, Czarnetzki, Brinkmann and Mussenbrock described it in 2008 under the name electrical asymmetry effect: it does not ask anybody to rebuild a chamber, only to change what the generator puts out.

Energy moved without moving the flux #

The point of the arrangement is what it does not change. The harmonic amplitudes are the same everywhere along the drawing’s horizontal axis, so the current the generator drives through the discharge, and the power it deposits, stay nearly fixed as the phase turns. The plasma’s density — and with it the flux of ions onto each surface — is set by that power. The energy each ion carries is set by the mean sheath voltage, and that is what the phase moves: from 80 electronvolts at the driven electrode and 36 at the grounded one, through equality at 45°, to the reverse at 90°.

In a single-frequency discharge the two cannot be separated. Raising the ion energy means raising the voltage, which raises the power and the density with it, so a process that needs gentle ions at high flux — etching a delicate layer quickly without damaging what lies underneath — has to compromise. Industry’s first answer was two unrelated frequencies, a high one to make the plasma and a low one to accelerate the ions, which works but couples the two through the heating of electrons at the sheath edge. Locking the second frequency to the first and choosing their phase gives a knob that acts almost entirely on energy. The generator’s settings for the two amplitudes fix the power; its setting for the phase fixes where the extremes fall; and because those are different properties of one waveform, the two controls barely interfere.

The separation is not perfect, and the reason is worth stating. Moving the bias moves the sheaths, which changes where and how strongly electrons are heated at their edges, which changes the density profile a little and with it ε. Simulations and measurements find the phase control of the bias somewhat weaker than the ideal line and the flux varying by tens of per cent across the phase range, rather than not at all. The model in the drawings, with ε fixed, is the first approximation to a loop that closes on itself.

More harmonics, sharper peaks #

Two harmonics give a bias of up to 0.44 of one harmonic’s amplitude. More harmonics give more, and there is a particularly clean family. Build a waveform from the first N harmonics of one frequency, all in phase, with amplitudes falling linearly from the fundamental’s — N, N − 1, down to 1, in units of the fundamental’s amplitude over N. As N grows the waveform becomes a train of narrow positive spikes separated by long, nearly flat negative stretches: the shape a sharp feature has to pay for in bandwidth, bought here with harmonics.

Scaled to a fixed peak-to-peak voltage, its maximum and minimum approach the ratio N to 1, and the clamp formula gives a bias of −(N − 1)/(N + 1) of half the peak-to-peak voltage: a third with two harmonics, three fifths with four, seven ninths with eight. The drawing checks every value against the formula. In the limit of many harmonics the driven electrode sits almost all the time at a steady negative voltage equal to nearly the whole swing, broken only by the brief spikes in which the sheath collapses to admit electrons. That is very nearly the ideal: a direct voltage applied to an electrode that cannot carry direct current, because it is behind an insulating wafer, with the electrons that neutralise the ions delivered in short pulses.

The narrow spikes have a second benefit that the single-frequency drive cannot give. Ions arriving at an electrode whose sheath voltage is constant for most of the cycle all fall through nearly the same potential, so the spread of their energies is small, and a process that selects a threshold — etching one material and not the one below it — can be run just above the threshold of one and below that of the other. Waveforms tailored this way, with peaks, with valleys, or with sawtooth ramps that make one sheath expand fast and the other slowly, are now studied as a family, and the number of harmonics a generator can deliver cleanly is the practical limit on how far the shape can be pushed.

Where the clamp stops being exact #

The sheath law. The square relation between a sheath’s charge and its voltage holds for a sheath of uniform ion density. A real collisionless sheath’s ion density falls towards the electrode as the ions speed up, and the voltage then goes as a different power of the charge; collisional sheaths differ again. The symmetric result, bias at the midpoint of the extremes, does not depend on the exponent, because it follows from the two sheaths being identical; the asymmetric formula and the detailed sheath waveforms do.

Collapse to exactly zero. A sheath does not fall to zero; it falls to a few electron temperatures, the floating value at which electron and ion currents balance for an instant. With electron temperatures of a few volts and harmonics of a hundred, that shifts the numbers by a few per cent.

The plasma as a wire. The plasma’s own inductance and resistance are neglected — the electrons’ reluctance to change their current, which in a wire is the circuit fighting its own change. In a strongly asymmetric discharge the sheaths’ nonlinear capacitance and the electrons’ inertia form a resonant circuit that rings at a frequency far above the drive — the plasma series resonance — and high harmonics in the drive can excite it. The electrons’ inertia is the same one that sets the frequency below which a plasma reflects everything, here showing up as a ringing in the discharge current. Tailored waveforms are designed with that in mind.

Collisions in the sheath. At higher pressures ions collide on their way across the sheath and arrive with a broad spread of energies well below the sheath voltage. The phase still moves the bias; it no longer delivers a narrow energy peak.

What the voltages do not show #

The drawings are one-dimensional: two plane electrodes and the plasma between. On a wafer thirty centimetres across driven at high frequency, the electrode is a sizeable fraction of a wavelength of the radio wave travelling across it, and the voltage is not the same at the centre and the edge; that standing-wave nonuniformity is one of the reasons industrial chambers stay at modest frequencies. Nor do the drawings show the ions’ angles, which decide whether an etched trench has straight walls. They show what a phase does to the direct voltage and the ion energy at a point, which is the part of the problem the clamp model answers exactly.

Still open: setting energy, flux and spread independently #

The two-harmonic scheme separates the ion energy from the ion flux reasonably well. A process engineer would like three independent controls — energy, flux and the width of the energy distribution — and possibly a fourth, the flux of energetic electrons that the collapsing sheath lets out, which can neutralise charge building up at the bottom of a deep trench. Whether a waveform of a handful of harmonics with chosen phases and amplitudes can deliver all of them at once, how many harmonics that needs, and how far the plasma’s own response to the waveform undoes the independence, are being worked out with simulations and with generators that deliver increasingly arbitrary waveforms, and the answer is not yet a design rule.

The habit worth carrying away is to ask what a nonlinear element responds to. A diode does not respond to a waveform’s average or its power; it responds to its extremes, so two drives with the same spectrum and different phases can produce different direct voltages. The sheath is a diode, the area law and the phase law are the same clamp seen through two different asymmetries, and a discharge that looks perfectly symmetric can still be told which way to lean.