Nothing keeps a magnetisation for ever

Every hysteresis loop so far has been drawn as though a magnetisation were something a material has. Sweep the field, read the magnetisation, and the loop says that the answer depends on the history — which is the whole content of remanence, and is why iron can be used to record anything.

It leaves out one variable, and the variable is time. A magnetised particle is not in its lowest energy state. It is in a local minimum, with a barrier between it and the reverse state, and the whole of its magnetisation depends on that barrier being too high to climb.

Barriers of finite height are crossed. Everything here follows from that one observation, and it is not a correction to the loop but a different account of what a loop is: the magnetisation of a real sample is a rate, the number measured depends on how long the measurement took, and there are particles for which the rate is fast enough that they have no remanence at all.

Louis Néel worked this out in 1949, and the consequences reached from the age of rocks to the capacity of a disk drive.

From a barrier to a time #

The particle’s moment is not sitting still. It is precessing, and being buffeted by the lattice’s vibrations, and at intervals of roughly a nanosecond it makes an attempt at the barrier. The probability that an attempt succeeds is a Boltzmann factor, so the average time before it succeeds is an attempt time multiplied by the exponential of the barrier in units of $k_BT$.

The number forty appears throughout the magnetic recording literature as the stability criterion a grain must meet, and it is worth seeing where it comes from, because it is neither measured nor conventional. It is $\ln(10\ \text{years}/1\ \text{nanosecond})$, and nothing more. Wanting a century instead of a decade would make it 42.6; accepting a year would make it 38.

What makes so crude a criterion useful is the exponential on the other side of it. A barrier ten per cent below the ten-year value gives a lifetime of two months; ten per cent above it gives four thousand years. The transition from “keeps it for ever” to “loses it at once” occupies about fifteen units of barrier, which at fixed anisotropy is a factor of 1.6 in the particle’s volume — a change of seventeen per cent in its diameter. There is no gradual regime worth speaking of.

That sharpness is why the phenomenon looks like a threshold when it is measured. A powder whose particles span a factor of two in diameter contains some that are utterly stable and some that are not, and nothing in between.

The size at which a magnet stops being one #

Fix the criterion and the anisotropy of the material and there is a volume, and from a volume a diameter.

A particle of iron nineteen nanometres across — some sixty thousand atoms — has a magnetisation that survives a decade. The same particle at fifteen nanometres has one that survives a second, and a powder of them behaves like a paramagnet whose individual moments happen to be ten thousand Bohr magnetons each. That state has its own name, superparamagnetism, and it is genuinely paramagnetic: no hysteresis, no remanence, a magnetisation curve that is a Langevin function of the field over the temperature, and a susceptibility enormously larger than any ordinary paramagnet’s.

It is also the reason such particles are useful. A superparamagnetic iron oxide particle can be pulled about by a field gradient and leaves nothing magnetised behind it when the field goes, which is why they are injected as contrast agents and used to separate cells. Note what has not changed at the threshold: the exchange interaction that aligns the moments inside the grain is untouched, and every one of the sixty thousand atoms is as firmly parallel to its neighbours as ever. What has been lost is the grain’s ability to hold that alignment pointing in one direction in the laboratory, which is a much weaker thing and is held by a much weaker energy. Superparamagnetism is a failure of anisotropy and not of ferromagnetism.

The problem this arithmetic set an industry #

The figure carries one point marked differently from the others, and it is there because the constraint it represents stopped a technology.

A bit on a magnetic disk is written across a few hundred grains, because the signal-to-noise ratio of a read-back goes as the square root of the number of grains in a bit and a few hundred is what is needed. Making the bits smaller therefore means making the grains smaller, and by the early 2000s the grains in a cobalt–chromium–platinum medium were approaching the diameter above, at which point they would begin to forget.

The obvious answer is to raise the anisotropy, which shrinks the stable diameter. But the same anisotropy sets the field needed to write the grain, and a write head can produce about 1.5 tesla and no more, because that is set by the saturation magnetisation of the best available soft magnetic alloy and there is no better one. So a medium stable enough to be small enough is a medium too hard to write.

That is the recording trilemma — signal, stability, writeability, pick two — and it is three consequences of one constant. Its resolution was to break the third requirement rather than the first two: heat the grain with a laser while writing it, so that its anisotropy falls, and let it cool in the field. Anisotropy falls much faster than magnetisation on approach to the Curie point, which makes the trick work; the engineering took twenty years.

The temperature at which a sample starts remembering #

The same statement can be made with the temperature as the variable instead of the size, and in that form it produced an instrument.

Fix a particle and lower its temperature. The barrier in thermal units rises, the lifetime rises exponentially with it, and at some temperature the lifetime passes the length of the measurement. Above that temperature the sample has no remanence and no hysteresis; below it, it has both. The crossing is called the blocking temperature, and the word is chosen well: nothing happens to the material there, and what changes is the relation between two times.

Which means the blocking temperature is a property of the experiment as much as of the sample. A magnetometer that averages for a hundred seconds and a Mössbauer measurement whose own timescale is a hundred nanoseconds assign blocking temperatures to the same powder that differ by a factor of two or more, and both are correct. Comparing two published blocking temperatures without checking the two measurement times is one of the standard ways of getting a wrong answer in this subject.

It is also how the subject became a dating technique. A lava flow cools through the blocking temperature of its magnetite grains in hours or days, and below it the direction each grain happened to have is frozen — so the rock records the Earth’s field at the moment it cooled, and the striped pattern on the ocean floor is a sequence of such records. Reading it requires knowing that the blocking temperature during cooling, at a rate of degrees per day, is not the blocking temperature in a laboratory measurement lasting a minute; the correction is a logarithm and it is a few tens of kelvin, which is small enough to be ignorable and large enough to matter for the hardest grains.

The general shape of it — a state that is stable only because the time to leave it is long — is the same as a supercooled liquid that has not found its crystal, and the escape statistics are the same ones a barrier crossing has anywhere. What is unusual about the magnetic case is that the barrier can be tuned from outside, continuously and reversibly, by applying a field — which is what the hero figure shows and what makes the next section possible.

A coercivity with a time attached #

If a barrier can be crossed by waiting, then a field slightly too small to remove the barrier will still reverse the particle — given time. So the field at which a sample switches depends on how long the field is applied, and the dependence is not slight.

Two facts follow that are easy to state and are routinely got wrong in practice.

The first is that a coercivity quoted without a measurement time is incomplete in the same way a velocity quoted without a frame is. A vibrating-sample magnetometer sweeping in three minutes and a pulsed-field magnetometer sweeping in a millisecond measure coercivities differing by ten or fifteen per cent on the same powder, and neither is in error; the material does not have one number.

The second is a design consequence and it is the more surprising. The figure’s curve falls steeply at short times and flattens at long ones, which means a field that writes a bit reliably in a nanosecond is a field that would erase it if left on for a year. Writing is possible precisely because it is done quickly, and data retention depends on the stray fields a disk experiences afterwards being both small and, crucially, not much smaller than the ones that would matter — since the barrier falls as the square of the field, a stray field of a tenth of the anisotropy field costs nearly a fifth of the barrier and a factor of a thousand in lifetime.

The drift that has no timescale #

Put a sample in its remanent state, leave it alone, and watch. It does not stay where it was put.

A logarithm is what a sum of exponentials looks like when their time constants are spread over decades: at every instant some population is in the middle of relaxing, and each decade of waiting brings a new population into play. So the drift is not a relaxation with a slow time constant. It is the absence of any time constant, and that is a stronger statement than it sounds.

The uncomfortable consequence is that there is no settling time. A sample that has drifted by one per cent in the first hour will drift by about another one per cent between an hour and ten hours, and again between ten and a hundred, and there is nothing in the physics that stops it — only the eventual exhaustion of the distribution. Anybody who has calibrated a permanent magnet has met this: a magnet knocked, heated or simply made behaves for months as though it were still deciding, and the standard treatment is to age it deliberately — heat it briefly, which drags the sample down the logarithm quickly, so that what remains drifts more slowly than the apparatus can detect.

The slope of the line is called the magnetic viscosity, and its name is misleading in the way technical names usually are: nothing is flowing and nothing is viscous. What is measured is the width of the barrier distribution near the point where it is being crossed, which is a statement about microstructure, and the same measurement is used as a probe of it.

Where the barriers come from #

The distribution the last figure needs is not an assumption. A real sample supplies it three times over, and the sources are worth separating because they respond to different things.

Sizes. The barrier is proportional to the volume, and a powder or a sputtered film has a distribution of grain sizes that is roughly lognormal with a width of twenty to thirty per cent. Cubed, that is a barrier distribution of order the mean.

Orientations. The astroid figure above shows the switching field varying by a factor of two with the angle between the easy axis and the field, and the barrier varies with it. A randomly oriented powder therefore has a wide distribution of barriers even if every grain is identical.

Defects. In a material large enough to contain a wall, the barrier is not an anisotropy energy at all but the energy of a wall getting past whatever is pinning it, which varies from pin to pin across a range that is essentially unbounded at the small end. That is the same population the crackle of a magnetisation curve is made of, seen on a different axis: the avalanches are what happens when the field is swept fast enough that the barriers are removed rather than climbed, and the creep is what the same barriers do when the field is held still.

All three are present in most samples. Which one dominates can be told apart by what changes the viscosity: annealing changes the defects, sieving changes the sizes, aligning the powder in a field during pressing changes the orientations.

What coherent rotation leaves out #

Everything here treats the particle as uniformly magnetised throughout the reversal. Real particles above about ten nanometres reverse through a non-uniform state — the magnetisation curls, or a reversed region nucleates at a corner — and the barrier for that path is lower, often much lower, than the coherent-rotation barrier the figures compute. Measured coercivities fall short of the anisotropy field by factors of three to five for exactly this reason, which is Brown’s paradox and is unresolved.

The attempt time is taken as one nanosecond and is not a constant. It depends on the damping, on the barrier itself and on the temperature, and published values range over three decades. Because it enters only inside a logarithm, a factor of a thousand moves the stability criterion by seven units out of forty, so the arithmetic survives — but no statement here should be read to more than two significant figures.

The anisotropy is taken as temperature-independent within each figure, and it is strongly temperature-dependent. Uniaxial anisotropy falls roughly as the tenth power of the reduced magnetisation, so a grain that is stable at room temperature may be superparamagnetic at 400 K, and the barrier in the lifetime figure is a function of $T$ as well as being divided by it. Every quoted diameter is a room-temperature diameter.

And the barrier distribution in the creep figure is Gaussian, which is a choice. Real distributions are closer to lognormal in the volume and are truncated by the sample’s own microstructure. What the figure establishes is that a broad distribution gives a logarithm and a narrow one does not; it does not establish that any particular sample’s distribution has the shape drawn.

A barrier crossed around the maximum rather than over it #

The landscape figure draws the energy against one angle, and a real particle’s moment moves in three dimensions. The path it actually takes over the barrier is not through the drawn maximum but around it, through a saddle, and for anything but a perfectly uniaxial particle the saddle is somewhere the one-dimensional picture does not contain. The barrier height happens to be right for the uniaxial case and the picture of how the reversal happens is not.

The lifetime figure draws an average and gives no sense of the distribution around it. Reversal is a Poisson process: a particle with a ten-year mean lifetime has a substantial chance of reversing in the first year and a substantial chance of lasting thirty. A disk drive’s specification is a statement about a failure probability across an enormous number of grains, not about a lifetime, and the two differ by more than a factor.

And none of these figures can show the thing a reader most wants to see, which is a grain in the act of forgetting. The reversal itself takes a nanosecond or so, and the waiting takes ten years; the ratio between them is the exponential that the whole subject is about, and no drawing can hold both ends of it.

There is a related omission in the creep figure that is worth naming because it is the one a reader is most likely to fill in wrongly. The curve is smooth, and what a sample actually does is jump: a grain reverses, or it does not, and the magnetisation of a sample of a million grains falls in a million discrete steps at unpredictable times. The smooth line is an ensemble average, in the same sense the exponential decay of a radioactive sample is an average over nuclei with no clocks in them, and a small enough sample watched closely enough shows the steps instead.

Still open: what the barriers actually are in a good permanent magnet #

The arithmetic above is clean for a single-domain particle whose barrier is its own anisotropy energy. A commercial neodymium magnet is not that. Its grains are five to ten micrometres across — thousands of times the single-domain diameter — so each of them can contain walls, and its coercivity is nevertheless a substantial fraction of the anisotropy field rather than the small fraction wall motion would allow.

What holds it is generally agreed to be nucleation: the grains are magnetised uniformly not because they are too small to divide but because nothing has started the division, and the coercivity is the field at which a reversed region can nucleate somewhere in the grain and grow. Where nucleation happens, what the local anisotropy is at that place, and why a thin grain-boundary phase of the right composition raises the coercivity by a factor of three, are questions answered at present by empirical metallurgy rather than by calculation. The temperature dependence of the coercivity is the main evidence, and it is consistent with a nucleation barrier of a few tens of thermal energies — which is to say that the arithmetic of this essay applies, with a barrier nobody can compute.

The habit worth carrying away is the one the coercivity figure forces. Before quoting a threshold, ask what measurement time is hidden in it. A coercivity, a yield stress, a breakdown voltage, a fatigue limit: each is the point at which something happens within the time somebody was watching, each falls as the observation is lengthened, and each is quoted as though it were a property of a substance. Where the underlying process is thermally activated, the dependence is logarithmic and therefore weak — which is exactly what makes it easy to ignore for a century and impossible to ignore once the application needs a decade instead of an afternoon.