The bias no battery supplies

The wall a plasma builds against itself found that any surface left alone in a plasma floats a few electron temperatures negative, because electrons reach it far faster than ions and it charges until the two arrivals balance. That is a statement about a wall that nothing drives. Almost every plasma that does useful work has a driven wall in it — a wafer on an electrode, a target being sputtered, a probe being swept — and most of them are driven at radio frequency, through a capacitor, by a waveform that is perfectly symmetric about zero.

The result is a large, steady, negative voltage that nothing supplies. The electrode of an etching tool — a surface that, left alone, would screen itself off from the plasma within a few Debye lengths and float a dozen volts down — sits a hundred or several hundred volts below the plasma, holds there for hours, and there is no direct current path anywhere in the circuit by which a battery could have put it there. The voltage is made by the plasma, out of the same asymmetry that floats an idle wall, and the rest of this essay is how.

A capacitor that can only be charged one way #

The sheath in front of an electrode is not a resistor. The current it passes depends on the electrode’s potential in a way one curve, three measurements already drew: a small, fixed ion current whatever the voltage, and an electron current that grows exponentially as the electrode approaches the plasma potential — the Boltzmann factor, the exponential that decides everything, applied to electrons climbing a potential hill. In argon, the electron current at the plasma potential is 108 times the ion current.

Put a capacitor between that electrode and a generator, and switch the generator on.

On the first positive half-cycle the electrode is pushed above the plasma potential and electrons pour onto it at the saturation rate. On the negative half-cycle the electrode is far below the plasma potential; electrons are turned back and only ions arrive, at a hundredth of the rate. The charge delivered in the positive half-cycle cannot be taken away in the negative one, so the capacitor keeps it. The electrode’s whole waveform shifts downward, the next positive peak reaches a little less high, admits a little less charge, and the shift continues.

It stops when the positive peaks only just reach the plasma potential — close enough that the burst of electrons each one admits equals the ions that arrive during the whole of the rest of the cycle. For a drive of 20 electron temperatures the electrode settles, within six cycles, at a mean of −22.3. The drive has not changed and is still symmetric. The asymmetry is entirely in what the plasma will let onto the electrode.

There is a circuit that does exactly this, and naming it makes the plasma less mysterious rather than more. A diode in parallel with the output of a capacitor-coupled signal is called a clamp, or a DC restorer: the diode conducts only at one extreme of the waveform, charges the capacitor on each peak, and holds the whole waveform on one side of zero. Television receivers used one to put back the black level that a capacitor-coupled video signal had lost. The sheath is that diode. Its forward current is the electron current, its tiny reverse current is the ion current, and its characteristic is the Langmuir probe curve. A plasma electrode driven through a capacitor is a clamp circuit in which the diode is made of plasma.

One cycle’s ions, one burst of electrons #

The steady state has a condition that decides the bias, and it is visible as two areas.

A capacitor in series passes no net charge in the steady state. Whatever charge the ions deliver over a cycle, the electrons must deliver the same amount of the opposite sign, and since the ions arrive steadily while the electrons are admitted only near the peak, the electrons have to do it in a hurry. At a drive of 20 electron temperatures the burst peaks at eleven times the ion current and is over within a sixth of the cycle.

That condition can be written down exactly. The electron current is the saturation current times $e^{V/kT_e}$, where the electrode potential is the bias plus the drive, $V = \bar V + V_0\sin\omega t$. Averaging the exponential over a cycle is a standard integral, and it gives a modified Bessel function:

$$1 = \sqrt{\frac{M}{2\pi m}}\; e^{\bar V/kT_e}\; I_0\!\left(\frac{V_0}{kT_e}\right).$$

Taking logarithms, the bias is the undriven floating potential minus $kT_e \ln I_0(V_0/kT_e)$. Nothing about the capacitor enters, provided it is large enough to hold its charge through a cycle, and nothing about the generator enters except the amplitude.

The integration behind the figures does not use that formula. It charges the capacitor step by step through thirty cycles with the ion and electron currents written from their own definitions, and the mean it settles at is then compared with the Bessel balance. They agree to the precision the step allows, which is the statement that the steady state and the balance are one fact.

The bias is the amplitude #

For a small drive the electrode barely moves from where it would float: a drive of one electron temperature shifts the bias from 4.68 to 4.92. For a large drive the Bessel function grows as $e^{x}/\sqrt{2\pi x}$ and its logarithm is very nearly the amplitude itself, so the bias approaches the drive amplitude, plus the floating potential, minus a slowly growing logarithm. At 10 electron temperatures the bias is 12.6; at 30 it is 32.1; at 100 it is 101.5.

In an argon discharge at 3 eV — ordinary for an etching tool — a drive of 100 volts gives a self-bias of −106 volts. A large enough drive clamps the electrode so that its most positive excursion just touches the plasma potential, which puts its average almost exactly one amplitude below. The few extra electron temperatures are the price of admitting enough electrons in a short enough burst.

That is the entire reason a radio-frequency supply can do what a direct-current supply cannot. A wafer is usually an insulator, or sits on one, and no direct current can be driven through it. A capacitor-coupled drive puts a steady voltage across the sheath above it all the same, and the insulator is simply part of the capacitor. The same arrangement lets an insulating target be sputtered: its surface charges negative by self-bias, ions are accelerated into it, and a material that could never carry a direct current is eroded atom by atom.

The effect also contaminates the instrument the whole theory of the steady sheath is read off. A Langmuir probe in a radio-frequency discharge sees the plasma potential oscillating, and its own sheath rectifies that oscillation exactly as a driven electrode rectifies its drive: the probe charges negative, its apparent floating potential drops, and its electron-temperature reading is smeared. Probes used in such plasmas carry inductors or driven compensation electrodes whose whole purpose is to cancel the self-bias this essay describes.

Two currents through one sheath #

The clamp picture counts particles arriving, and at the frequencies tools use, most of the current through a sheath is not particles at all. A sheath a millimetre thick is a capacitor whose plates are the electrode and the plasma, and a voltage swinging by a hundred volts at 13.56 MHz drives a displacement current through it — the term Maxwell added to Ampère’s law, the one the term that made light is about — of about 7.5 milliamperes per square centimetre. The ion current an argon plasma at $10^{16}$ per cubic metre and 3 eV delivers to the same electrode is about a quarter of a milliampere per square centimetre, thirty times smaller.

So at radio frequency the sheath is mostly capacitor and only slightly diode. That does not undo the self-bias, and the reason is worth being precise about. The displacement current averages to zero over every cycle, because it is the rate of change of a charge that returns to where it started. The conduction currents average to zero only if the bias makes them, and the steady voltage on the blocking capacitor is set by exactly what does not average away. The capacitive current decides the shape of the voltage across the sheath through a cycle; the conduction current decides where its average sits. The Bessel balance is a statement about the second, and it survives any amount of the first, provided the electrode still touches the plasma potential once a cycle to admit its electrons.

This is the division the bill that arrives when the pushing stops draws for a single radiating charge: a reactive part, exchanged back and forth every cycle and settling nothing, and a real part that does not come back and decides the steady state. It is also why frequency separates the two species so cleanly. Electrons follow fields oscillating well below the frequency below which nothing gets in — the electron plasma frequency, close to a gigahertz in these discharges — and track a 13.56 MHz drive instantly. Ions follow only below the ion plasma frequency, about 3 MHz in argon at this density, and above it they feel an average. A tool driven between the two has, by construction, electrons that clamp and ions that do not.

Two electrodes, and which one takes the voltage #

A real chamber has two surfaces the discharge current flows between: the powered electrode, and everything grounded — usually the chamber walls, much larger. The plasma floats between them, and the question is how the steady voltage divides across the two sheaths.

At the frequencies tools use, the sheaths are mostly capacitive: the current through them is displacement current, and it is the same current through both, since it flows from one electrode, through the plasma, to the other. The radio-frequency voltage therefore divides between them in inverse proportion to their capacitances, and a sheath’s capacitance is its area divided by its thickness. A small electrode has a small capacitance and takes most of the voltage.

The subtle part is that thickness depends on voltage. Child’s law, for a collisionless sheath holding a large voltage, makes the thickness grow as the three-quarter power of the voltage, so the small electrode’s sheath, taking more voltage, grows thicker, its capacitance falls further, and it takes more voltage still. The feedback converges — the figure iterates it to its fixed point — and the result is a voltage ratio equal to the area ratio raised to the fourth power. That fourth power, derived by Koenig and Maissel in 1970, is the number every textbook quotes: a wall five times the area of the electrode puts 625 times the voltage across the electrode’s sheath.

Real discharges do not reach it. Measured exponents lie between about one and two and a half, because ions collide while crossing the sheath (which changes the thickness law), because some of the current is conduction rather than displacement, and because the grounded surfaces are not a single plane. The fixed-thickness case gives exponent one and the collisional law gives two and a half, and they bracket what is measured. The direction is the same in all of them, and it is what the design of every etching chamber follows: the wafer goes on the small powered electrode, and the chamber is made large and grounded, so that nearly all the bias appears where the ions are wanted and almost none appears at the walls, which would otherwise be sputtered onto the wafer.

What arrives at the wafer #

The self-bias is an average, and an ion does not strike the wafer with an average. It strikes with whatever energy the sheath gave it during its own crossing, and the sheath voltage is swinging through a full cycle while the ion crosses.

Two limits bracket the answer. An ion that crosses in a small fraction of a cycle sees the instantaneous sheath voltage and lands with it. Since a sinusoid spends most of its time near its extremes, the distribution has two sharp peaks, one near the lowest sheath voltage and one near the highest, separated by twice the amplitude — 79.7 electron temperatures at a drive of 40. An ion that takes many cycles to cross sees the voltage averaged over its transit and lands near the mean, and the spread collapses.

In between, the spread is the swing multiplied by $\sin(\pi\tau)/\pi\tau$, where τ is the transit time in cycles, which is the transfer function of a moving average — the same function sharpness has to be paid for finds as the Fourier transform of a rectangular window. Each ion is a low-pass filter applied to the sheath voltage, and its transit time is the width of the window. The figure’s spreads follow that function to within the extra harmonics of the real sheath waveform: 60.5 at 0.4 cycles, 15.8 at 1.3, 6.2 at 3.3.

Heavy ions respond to a rapidly varying field by feeling only its average, and that is the principle held up by a force that averages to nothing uses to explain the Paul trap and the inverted pendulum. Here it appears as a design parameter. Argon crossing a millimetre-thick sheath at a hundred-odd volts takes about 120 nanoseconds. At 1 MHz that is a tenth of a cycle and the ions arrive with two energies; at 60 MHz it is seven cycles and they arrive with nearly one. The frequency of an etching tool is chosen for its heaviest particles, because the electrons follow any frequency anyone uses and the ions do not.

The frequency most tools actually use, 13.56 MHz, was not chosen for any of this. It is an industrial, scientific and medical band in the international radio regulations, reserved so that equipment radiating at high power there does not interfere with communications, and the physics was fitted to the regulation rather than the other way round. At that frequency the argon transit is 1.65 cycles, squarely in the intermediate range where neither limit applies — which is part of why modern tools drive at two frequencies at once, a low one to set the ion energy and a high one to sustain the plasma.

Where the clamp model stops being right #

The self-bias balance treats the sheath as a conductor. It counts ion and electron conduction currents and ignores the sheath’s displacement current, which is the right accounting for the average charge on the capacitor and the wrong one for the waveform across the sheath at high frequency. The bias equation holds while the electron burst still reaches the electrode in each cycle; the area law holds in the opposite, capacitive limit. Both are limits of one problem that a full treatment solves together.

The plasma potential is held fixed. In a symmetric chamber, or one whose grounded area is not much larger than the powered one, the plasma potential oscillates with the drive, and the electrode’s excursion relative to the plasma is less than its excursion relative to ground. The single-electrode figures describe a powered electrode facing a very large grounded wall.

The electrons are Maxwellian and the ion current is fixed. Radio-frequency discharges heat their electrons at the moving sheath edge, which produces distributions that are not thermal, and the Bohm ion current depends on the electron temperature at the sheath edge, which changes through the cycle. Both change the numbers of the balance and neither changes its structure.

Transit is modelled as an average over a window. A real ion is accelerated through a sheath whose thickness is also oscillating, so its transit time depends on when it entered, and ions that collide with neutral gas on the way lose the two-peaked structure entirely and arrive with a broad tail. The figure’s rows are the collisionless idealisation.

And the areas are effective areas. Which surfaces count as grounded depends on where the plasma actually touches, which in turn depends on pressure and power; the measured exponents between one and two and a half partly reflect how uncertain the effective area of a chamber wall is.

What the traces leave out #

The figures are all functions of time or of a ratio, and the thing that is actually happening is a moving boundary. The sheath in front of the electrode expands to several millimetres when the electrode is at its most negative and collapses almost to nothing when it touches the plasma potential, once every cycle, and the electrons in its path are pushed back into the plasma and heated as it advances — a collisionless exchange between a moving field and the particles that happen to be in phase with it, of the same character as the wave that dies with nothing to rub against. That motion is where most of the power in a capacitively coupled discharge goes, and none of the traces can show it, because it is a thickness changing in space rather than a voltage changing in time.

Nor do they show the chamber. The self-bias of a real tool depends on its geometry in ways the two-area model compresses into one number, and the uniformity of the bias across a 300-millimetre wafer — the property that decides whether the chips at its edge work — is a two-dimensional problem the figures do not approach.

Still open: making the bias without the geometry #

The area law makes self-bias a property of the chamber, which is inconvenient: a tool built with a given ratio of areas has its ion energy and its plasma density tied together, because both follow from the same drive. For two decades the answer was to add a second, lower frequency and hope the two did not interact, and they do interact, through the electron heating at the sheath edge that both frequencies share.

A different answer has been developed since 2008. Driving a geometrically symmetric chamber with a fundamental frequency and its second harmonic, with a controlled phase between them, produces a waveform whose positive and negative excursions differ, and that asymmetry in the waveform does what an asymmetry in the areas used to do: it generates a self-bias, adjustable continuously by the phase alone. Whether waveforms tailored in this way can give independent control of ion energy, ion flux and the energy spread at the same time — and how many harmonics that takes — is an active question in the design of the next generation of tools.

The habit worth carrying away is to look for the rectifier. Wherever a response is exponential on one side and flat on the other, an alternating drive produces a steady offset, and the offset is set by the asymmetry of the response rather than by anything in the drive. A plasma sheath, a diode, a crystal radio and an ion drifting through an oscillating field all do this, and the size of the effect is always the size of the drive.