The first length that belongs to the substance

Three lengths have come out of this subject already, and every one of them has been a length of the object rather than of the material.

A magnet’s demagnetising factor is a pure number computed from its proportions. The largest avalanche a magnetisation curve contains is cut off by the sample’s own restoring field, which is the same factor again. Both are facts about a piece rather than about a metal, and both change if the piece is cut in half.

This essay is about a length that does not. It is the width of the boundary between two domains, it is made of two material constants and nothing else, and it is where the subject acquires a scale of its own.

Two energies with opposite slopes #

Inside a domain the moments are parallel, held that way by an exchange interaction thousands of times stronger than anything magnetic — an interaction that is not magnetic at all, and which classical statistical mechanics cannot even contain. Between two domains pointing opposite ways, the moments have to turn through half a revolution somehow.

The obvious arrangement — one plane of atoms pointing up and the next pointing down — is the most expensive one available. Exchange penalises the angle between neighbours, and it penalises it quadratically: the cost of a bond at angle $\theta$ goes as $1 - \cos\theta \approx \theta^2/2$. So turning through $\pi$ in one step across one plane costs the full exchange energy of every bond crossing it.

Spread the same turn over $N$ planes and each step is $\pi/N$. Each bond then costs $\pi^2/2N^2$ of the exchange energy, and there are $N$ planes paying it, so the total cost falls as $1/N$. The quadratic penalty is not an approximation chosen for convenience: it is the leading behaviour of any energy at a minimum, which is why so much of physics is a parabola, and it is what makes spreading a distortion out so cheap. A turn spread over a hundred atoms costs a hundredth of an abrupt one, and the exchange interaction alone would spread the wall across the whole crystal.

Something stops it. The crystal has easy directions — a cubic crystal of iron is easier to magnetise along a cube edge than along a body diagonal, by an energy density $K$ that is a property of the lattice rather than of the shape — and every atom inside the wall is pointing somewhere that is not an easy direction. That cost is paid per unit volume, so it rises in proportion to the width.

The minimum is at $w \sim \sqrt{A/K}$ and the cost there is $\sim\sqrt{AK}$, where $A$ is the exchange stiffness — the exchange energy per unit length of the crystal, about 21 picojoules per metre for iron — and $K$ is the anisotropy energy density.

Two things are worth noticing about that result before any numbers are put into it. Neither constant contains a dimension of the sample, so neither does the answer: an iron wall in a pinhead and an iron wall in a girder are the same width. And the two quantities enter in opposite senses, so a material cannot have a narrow wall and a cheap one; that trade turns out to be the whole difference between a permanent magnet and a transformer core.

The wall the crystal actually has #

The scan above assumed the magnetisation turns at a uniform rate, which is a guess. Minimising the energy properly — treating the angle as a function of position and asking which function is cheapest — gives an equation for the profile and a closed solution for it.

The exact wall turns slowly at its edges and quickly in the middle, because the anisotropy cost is largest where the moment points furthest from an easy direction and it pays to cross that region fast. Its energy is $4\sqrt{AK}$ exactly — four millijoules per square metre for iron — and the eleven per cent saving over the uniform guess is the whole benefit of doing the variational problem properly.

It also has no edge. The solution approaches the two domain directions exponentially and never quite reaches them, so a wall has no definite thickness and what is quoted as its width is a convention: the width the tangent at the centre would need if the turn continued at that rate, which is $\pi\sqrt{A/K}$. That is not a defect of the calculation. It is what the answer is, and it is worth carrying because the same thing is true of every interface described by minimising an energy — the surface of a liquid, which is not a skin and has a thickness of a few molecules, the edge of a superconducting region, the front between two phases that a new phase has to nucleate.

The profile has a second property worth naming, because it puts the wall into a class with objects from elsewhere in physics. It is a localised solution of a nonlinear equation, it holds its shape, and it can be translated anywhere without cost — the equation contains no preferred position, so a wall shifted sideways is another exact solution of the same energy. That is the defining behaviour of a solitary wave, and the mathematics is literally the same: the equation $A\theta'' = K\sin\theta\cos\theta$ is the sine-Gordon equation, and its kink is this wall. The zero-energy state bound to a domain wall in a polymer chain is the same object again, in a different medium and with different consequences.

Iron’s sixty-six nanometres is about two hundred and fifty atomic planes. Set beside the exchange energy, that makes the wall extraordinarily cheap: the angle between neighbours inside it is about seven tenths of a degree, the quadratic penalty on that angle is seven parts in a hundred thousand, and the energy stored in the whole wall works out at roughly a twentieth of a kelvin per atom in it against an exchange energy of ninety kelvin. A domain wall is a structure that costs almost nothing, spread over a distance that costs almost nothing to spread over, which is why a piece of iron divides into domains as readily as it does.

Seven magnets, and the line that divides them #

With the two constants identified, the range they cover across real materials is the interesting quantity.

Permalloy’s wall is nine hundred and twenty-five nanometres wide and costs 0.18 millijoules per square metre. Samarium cobalt’s is two and a half nanometres and costs fifty-seven. The two materials are metals of similar atoms at similar spacings, and their exchange stiffnesses differ by ten per cent; what differs is the anisotropy, by five decades, and the wall reports it as a length.

The product of the two is exactly $4\pi A$ for every material on the figure, which the figure checks rather than takes on trust. Since $A$ varies by less than a factor of four across the whole set, there is essentially one number to trade: a material with a narrow wall has an expensive one and a material with a cheap wall has a wide one, and no choice of composition escapes it.

That trade is what the shaded band is about. A material can hold a magnetisation against its own demagnetising field only if its anisotropy exceeds the energy density of that field, which is the condition $\kappa = \sqrt{K/K_d} > 1$ with $K_d = \mu_0M_s^2/2$. Neodymium iron boron, ordered iron platinum and samarium cobalt are on the right of the band and are what permanent magnets are made of. Permalloy, nickel and iron are on the left and are what transformers and read heads are made of. Cobalt sits inside it, which is why cobalt alone makes an indifferent permanent magnet and an excellent ingredient in one.

So a division that looks like a catalogue of applications — hard materials here, soft materials there — is a single inequality between two energy densities, and the wall width is that inequality read as a length. A material whose wall is a few nanometres wide is a permanent magnet; one whose wall is a micrometre wide is not.

Why a narrow wall is a hard wall #

The connection between the wall’s width and the coercivity is worth being careful about, because the usual shorthand gets the mechanism backwards.

A wall moves when something pushes it past whatever is holding it. What holds it is a defect — a precipitate, a grain boundary, a region of different anisotropy — and how strongly a defect holds depends on how the defect’s size compares with the wall’s. A defect much smaller than the wall is averaged over and barely felt; a defect comparable to the wall changes the energy of the wall substantially as it passes. A narrow wall is sensitive to fine-grained disorder that a wide wall does not notice, which is one reason the same density of defects pins much harder in a hard material. It is also why the crackle a magnetisation curve makes is a probe of microstructure at all: the wall’s width is the length scale at which it samples the crystal, and a signal produced by an object nine hundred nanometres wide reports on quite different features from one produced by an object three nanometres wide.

But that is not the main reason a permanent magnet is hard, and the last figure of this essay is why.

The second length, and the only two there are #

The exchange length has appeared twice now without being named, and it is worth pausing on because magnetism has exactly two material lengths and this is the other one.

The wall width compares exchange with anisotropy. The exchange length compares exchange with the material’s own magnetostatic energy: $\sqrt{2A/\mu_0M_s^2}$, which is between two and six nanometres for everything on the figure above. It is the distance over which the exchange interaction can force the magnetisation to stay uniform against its own demagnetising field, and below it nothing can make the magnetisation vary — a structure smaller than the exchange length is magnetised uniformly whether that is favourable or not.

Those two lengths are made of three constants, and their ratio is the hardness parameter: $\kappa$ is the exchange length divided by the wall parameter, up to a factor of two. So the band on the figure above, the coercivity a material can reach, the size below which a particle stays uniform and the width of its walls are four readings of one comparison between an anisotropy energy density and a magnetostatic one.

A subject with two lengths in it has one dimensionless number, and almost everything in applied magnetism is a statement about where a material sits on it. That is a considerable compression of a field that looks, from its catalogues, like an unrelated collection of alloys.

The particle that cannot afford to divide #

A body divides into domains because dividing reduces the demagnetising energy, and it pays for that reduction with the area of the wall it has to create. The saving goes as the volume and the cost as the area, so below some size the arithmetic reverses.

A particle below its critical diameter has no domains. It has no wall, so it has nothing to pin and nothing to move, and the only way it can reverse is for every moment in it to rotate together against the anisotropy — which takes the full anisotropy field, $2K/\mu_0M_s$, and is enormous.

This is the whole design principle of a modern permanent magnet, and it is a statement about a length rather than about a material. Grind the material fine enough that each grain is below its critical diameter, and the coercivity stops being a question about defects and becomes a question about anisotropy. That is why neodymium magnets are sintered from powders of a few micrometres with a grain structure finer still, and why the coercivity of such a magnet falls when the grains are allowed to grow.

It is also where the estimate stops working, and the figure is drawn so as to say so. For permalloy the comparison returns four nanometres, which is smaller than permalloy’s own exchange length of five point seven — the distance over which the exchange interaction can be overcome at all. A wall cannot be narrower than that, so the wall whose cost the comparison priced does not exist at that size, and the number means nothing. Soft particles do have a single-domain size, of a few tens of nanometres, and it is set by the exchange length rather than by this crossing. A calculation that compares two things has to be checked for whether both of them exist.

The wall that goes slower when pushed harder #

One consequence of the wall being an object in its own right deserves stating, because it is the most surprising thing about it and it is a direct consequence of the width.

A wall in a field moves, and the moments inside it do not simply rotate: each one precesses about the field it finds itself in, and the precession tilts the magnetisation out of the plane of the wall. That tilt creates a demagnetising field, the moments precess about that, and the net result is a steady sideways translation. A wall therefore moves because it is precessing, not in spite of it, and its velocity is proportional to the applied field with a constant that involves the damping.

Up to a point. Push harder and the out-of-plane tilt grows, and past a tilt of forty-five degrees the demagnetising field it produces stops growing with it. The wall then cannot translate steadily at all: it begins to precess as a whole, turning inside out repeatedly, and its average velocity falls as the field is raised further. The threshold is the Walker field, it is of order the wall’s own demagnetising energy expressed as a field, and it is a few millitesla in permalloy — a very small field.

The consequence for anything that moves domain walls deliberately is severe: there is a maximum useful velocity, of order a hundred metres per second in a soft film, and driving harder is worse than useless. It is a good example of a limit that belongs to the internal structure of an object rather than to the force applied to it, in the same family as the chatter that a stiffer holder removes — where an outcome is decided by a stiffness rather than by the driving force — and it is invisible in any description that treats the wall as a boundary with a position and nothing else.

Where a Bloch wall is the wrong wall #

Every wall here is a Bloch wall in an infinite crystal. In a thin film the moments in a Bloch wall would have to point out of the plane at the centre, which the film’s own demagnetising field forbids, so below a thickness of order the exchange length the wall turns in the plane instead — a Néel wall, with a different profile, a different energy and a different width. Read heads and magnetic memory are made of exactly such films.

The anisotropy is taken as uniaxial and one constant. Cubic iron has a second constant and three easy axes, so its walls come in ninety-degree varieties as well as one-hundred-and-eighty-degree ones, with energies about a third as large. The figures use $K_1$ throughout, which sets the scale correctly and is not the whole of any real anisotropy.

The constants are room-temperature values and both depend strongly on temperature. Anisotropy falls much faster than magnetisation as the Curie point is approached — roughly as the tenth power of the reduced magnetisation for a uniaxial material — so a wall that is three nanometres wide at room temperature is far wider at 500 K. That dependence is not a nuisance; it is what heat-assisted magnetic recording exploits.

And the single-domain estimate assumes a sphere and a uniform magnetisation on either side. A real small particle reverses through a non-uniform state — a curling or vortex mode — at a field well below the coherent-rotation value, which is the standard explanation for why measured coercivities fall short of the anisotropy field by factors of three to five. That shortfall has a name, Brown’s paradox, and it has not been fully resolved.

One component of a vector that turns in three #

The profile figure plots one component of the magnetisation against one coordinate, and a wall is a rotation in three dimensions of a vector field in three dimensions. What it looks like is a helical twist: the moments turn in the plane of the wall, so the component drawn is a cosine of an angle whose other component is out of the page entirely. Nothing about the handedness of that twist — which is a real property, and which decides how a wall responds to a current passed through it — is visible in a plot of one component.

The materials figure draws seven points and cannot show that the constants behind them are not measurements of the same kind. A saturation magnetisation is known to a per cent; an exchange stiffness is inferred from spin-wave measurements or from a Curie temperature and is uncertain by twenty; an anisotropy constant depends on the state of the sample, its temperature and its strain, and quoted values for the same material differ by factors approaching two. The decades on that axis are real and the third significant figure is not.

Still open: whether a wall in a real crystal is anything like this one #

Every wall in this essay is a solution of a continuum problem — an energy written as an integral over a smooth field of magnetisation, minimised. That description cannot be right down to the atomic scale, and for the hardest materials the wall is only ten atomic planes wide, which is not obviously enough planes for a continuum to mean anything.

What happens to a wall when it is a few lattice spacings wide is a genuine question rather than a technicality. A continuum wall can sit anywhere; a wall that is a few atoms wide feels the lattice itself, and its energy depends periodically on where its centre is — an intrinsic pinning with a period of one lattice spacing, which exists in a perfect crystal with no defects at all. The barrier that produces is calculable in models and has been seen in the ordered alloys where walls are narrowest, and how much of the coercivity of a real hard magnet it accounts for is argued about.

The habit worth carrying away is the one the last figure forced. When a calculation compares two quantities, check that each of them exists in the regime where the comparison is being made. The single-domain diameter is a competition between a wall’s cost and a division’s saving, and it is meaningless wherever the answer comes out smaller than the wall — which is precisely the soft materials, precisely the ones the formula is most often quoted for, and precisely the case where a plausible number is returned and nothing warns that it means nothing.