The polarisation nobody asked for

A crystal is polar when its class fixes a direction — when there is an axis no operation of the group moves — and a polar crystal can carry a built-in electric polarisation. This collection has counted the polar classes and takes the count as given, along with the distinction between permission and presence that runs through every property argument here.

The interesting case is how a crystal becomes polar. The obvious route is that the polarisation is the order parameter: the crystal’s atoms separate their charges, the polarisation grows from zero, and the symmetry drops because the polarisation is there. That is a proper ferroelectric.

The other route is stranger. Something else entirely is the order parameter — an ordering, a rotation of a group of atoms, an alternation from cell to cell — and the phase it produces happens to be one whose class permits a polarisation. Then a polarisation appears, small, as a side effect, in a crystal whose transition was about something else. That is an improper ferroelectric, and this essay computes which of the plane’s transitions are of that kind.

Two questions that are not the same #

The distinction rests on separating two questions that a casual reading runs together.

Does the mode carry a dipole? That is a fact about the displacement pattern: sum the displacements over the cell, weighted by the charges of the atoms, and see whether the result is zero. It is a first-order quantity, linear in the order parameter’s amplitude.

Can the resulting phase hold a polarisation? That is a fact about the group the frozen structure has: does its class fix a direction? It is a yes-or-no question about a subgroup, with no amplitude in it at all.

For a proper ferroelectric both answers are yes, and they are the same fact. For an improper one the first is no and the second is yes, and the gap between them is where the physics lives: a polarisation the mode does not produce at first order can still appear at second, because the energy contains a term coupling the polarisation to the square of the order parameter.

The coupling, as a character count #

Whether such a term exists is a character computation and this ladder has the machinery for it.

The energy may contain a term linear in the polarisation and quadratic in the order parameter exactly when the symmetric square of the order parameter’s representation contains the vector representation. That multiplicity is an inner product of characters, computed in exact cyclotomic arithmetic, and it is an integer.

Where it is non-zero, the polarisation is forced to follow the square of the order parameter: it is zero above the transition, grows as the square below it, and its size is fixed by a coupling constant nobody can compute from symmetry. Where it is zero, no such term exists and the polarisation stays zero to that order.

Both halves are computed here and compared. The coupling count says a polarisation should appear; the isotropy subgroup says whether the phase can hold one. A coupling predicting a polarisation in a phase whose class fixes no direction would mean one of the two computations was wrong, and that consistency check runs over every mode of every group.

The zone-centre result, which is a negative #

Running the enumeration at the zone centre in the plane gives a clean and slightly disappointing answer: there are no improper cases at all.

The reason is structural rather than accidental. At the zone centre the order parameters available are the one-dimensional characters and the natural two-dimensional representation. For a one-dimensional order parameter the symmetric square is trivial, so the coupling term exists only if the parent already had a polar direction — in which case the parent was already polar and nothing improper has happened. And the two-dimensional order parameter, in the plane, is the vector representation, so a transition driven by it is a proper ferroelectric by definition.

So the plane’s zone centre has nothing to offer. That is a computed negative, and this collection’s standing position is that a computed negative is a result: it says where to look rather than closing the subject.

Where to look instead: the zone boundary #

At a zone-boundary wavevector the situation is different, and for a reason that can be stated in one line: a mode that alternates from cell to cell has a dipole of exactly zero, because the contributions of neighbouring cells cancel.

That is not an approximation. The displacements in adjacent cells are exact negatives of each other, so the sum over the doubled cell is zero as an exact rational, and the machinery reports it as such rather than as something small.

Meanwhile the phase such a mode produces can perfectly well have a polar class. The mode has removed operations — including, often, every operation that would forbid a polar direction — and what is left may fix one.

So the improper cases in the plane are all at the zone boundary, and the enumeration finds them: modes of p3m1 and p31m at wavevectors doubling the cell, whose frozen structures have classes that permit a polarisation while the modes producing them carry none.

What the enumeration actually found #

The honest report is that the improper cases in this plane enumeration are thin, and the reasons are worth naming rather than hiding.

Of the hundred and eighty modes computed, the ones satisfying all three conditions — parent not polar, mode carrying no dipole, child’s class polar — number twenty-four, and all of them come from the three parents with a threefold axis and no sixfold — p3, p3m1 and p31m. Most produce a phase of class 1, which permits a polarisation in the trivial sense that a structure with no symmetry permits anything; four produce a phase of class m, which is a genuinely polar class with a mirror in it.

Those four are the plane’s improper ferroelectrics, such as they are. In three dimensions the same enumeration over the two hundred and thirty groups produces the cases the literature is about, and this collection has not run it — the two hundred and thirty groups would need the same machinery over a much larger enumeration.

The plane is a poor place for this phenomenon, and saying so is the useful conclusion. Improper ferroelectricity needs an order parameter that is not the polarisation and a phase that is polar anyway, and the plane’s small stock of representations makes that combination rare.

Why the threefold groups and no others #

The three parents supplying every improper case in the plane are p3, p3m1 and p31m, and their common feature is the one that matters.

A threefold axis is the only rotation the plane permits whose order is odd and greater than one. That single fact has produced three separate results in the essays around this one: the threefold groups are the ones with a cubic invariant, the ones whose invariant ring has an odd-degree generator, and now the ones supplying improper cases.

The link in this case runs through the zone-boundary wavevectors. Doubling a hexagonal cell in one or both directions produces a supercell whose own symmetry is low — an oblique lattice — so the operations surviving a mode there are few, and the classes that result are small enough to be polar. A square parent doubled the same way keeps a rectangular supercell, whose surviving classes are 2mm and its subgroups, and 2mm is not polar.

So the availability of improper cases in the plane comes down to what the supercell permits, and the hexagonal lattice’s supercells permit more than the square lattice’s. That is a fact about lattices rather than about representations, and it is the sort of dependence a purely representation-theoretic account would not have predicted.

The order parameter that is not the polarisation #

The secondary quantity in an improper transition has a name in the literature — the secondary order parameter — and its behaviour is worth setting against the primary one, because the two are often confused in a description of a material.

The primary order parameter is what becomes non-zero first and what the transition is about; the secondary is anything the primary drags along through a coupling. A spontaneous strain is the commonest secondary parameter and appears in most of the descents this ladder computes; a polarisation is a secondary parameter in exactly the improper case.

There is no symmetry difference between them in kind — both are quantities that become non-zero — and the difference is entirely in the order of the coupling. A quantity transforming as the same representation as the primary parameter is dragged linearly and is really the same parameter; one transforming so that the coupling is quadratic is dragged as the square.

That is why the arithmetic here is a count of representations in a symmetric square rather than a comparison of representations directly. The question is not whether the polarisation transforms like the order parameter — it does not, or the case would be proper — but whether it transforms like something the order parameter’s square contains.

What a polar class actually says #

The condition on the phase is that its class fix a direction, and it is worth being exact about what that does and does not mean, because this collection has an essay on the gap.

A polar class permits a polarisation. It does not require one: a crystal in a polar class can perfectly well have no net dipole moment, and many do. And a permitted polarisation may be too small to measure or may be screened by charges the crystal picks up from its surroundings.

What the class forbids is the opposite case, and the prohibition is the useful half: a crystal whose class is not polar cannot carry a built-in polarisation at all, whatever its chemistry. Every one of the phases in the census with a non-polar class is excluded outright, and no coupling can produce a polarisation in them.

So the enumeration is best read as a filter rather than a prediction. Twenty-four modes leave a phase that could hold a polarisation; four of those leave a phase whose class is m rather than trivial; and the physics of whether any real material takes such a route is not in the arithmetic.

Why an improper polarisation is small, and useful #

The physics the arithmetic points at is worth a paragraph, stated as physics and not derived here.

Because the polarisation follows the square of the order parameter, it is small near the transition — smaller than a proper ferroelectric’s, which follows the parameter itself. It also responds differently to an applied field: reversing it means reversing the order parameter, which the field couples to only through the same quadratic term, so the coercive behaviour is not the proper case’s.

The compensating advantage is that the polarisation is tied to something structural rather than to a soft electrical mode, so it can be less sensitive to temperature near the transition and can survive in materials whose direct polar instability is absent. Those are the properties that make improper ferroelectrics interesting in practice, and none of them is computable from symmetry.

What symmetry supplies is the permission and the mechanism: the coupling term exists or does not, and the phase can hold a polarisation or cannot. Everything about size, response and usefulness is elsewhere.

The measurement that separates the two #

There is an experimental signature that distinguishes proper from improper, and it follows from the arithmetic above rather than from any new principle.

For a proper ferroelectric the polarisation is the order parameter, so it grows as the order parameter does — and any superlattice reflection associated with the transition, if there is one, grows as the square of the polarisation. For an improper one the relationship is inverted: the superlattice reflections grow as the square of the order parameter, and the polarisation grows as the square as well, so polarisation and superlattice intensity are proportional to one another.

That is a statement about two measurable quantities and their relative behaviour, and it needs no coupling constant to test. It also connects this essay to the superlattice reflections a cell-doubling transition produces: in the improper case, the reflections and the polarisation appear together, because both are consequences of the same order parameter.

Three ways this can be got wrong #

Reading the trivial class as polar. A phase of class 1 fixes every direction, so it trivially permits a polarisation — and a routine that counted such rows as improper ferroelectrics would report a great many. The count above separates them, and only the four with class m are genuine.

Forgetting that the parent may already be polar. If the high-symmetry phase has a polar class, the crystal was polar before the transition, and a polarisation in the low-symmetry phase is not news. The enumeration excludes those rows explicitly.

Taking a first-order dipole for a polarisation. The sum of the displacements is a dipole moment of the mode, computed with the sign of the orbit standing in for the charge. It is a model quantity, not a measured one, and its exactness is a fact about the model. What it establishes is whether the mode is polar by symmetry, which is what the argument needs — the same reasoning the mechanical representation uses to decide which modes a site can supply at all.

What is checked, and what is asserted #

Checked. That every mode’s frozen structure has the group its equivariance predicts, on all one hundred and eighty. That the dipole of every zone-boundary mode is exactly zero. That no mode carrying a dipole lands in a phase whose class cannot hold one — a consistency requirement between two computations that could disagree. And that the improper case occurs at all, which is the assertion that would fail if the enumeration were empty.

Asserted and not computed. That an improper polarisation grows as the square of the order parameter — that is Landau theory, imported. That its size is small — that depends on a coupling constant. That such materials exist — the literature’s examples are three-dimensional, and none of them is computed here.

And the boundary of the claim. This is a plane enumeration over seventeen groups at four wavevectors. The three-dimensional case has more representations, more wavevectors and more polar classes, and the phenomenon is correspondingly commoner there; nothing in this essay establishes anything about it beyond the mechanism.

The materials the plane cannot supply #

The enumeration’s conclusion is that the plane is a poor place for this phenomenon, and it is worth setting three-dimensional cases beside it, because they show the mechanism doing what the plane’s four cases only gesture at.

Gadolinium molybdate is the classical example, and it has been the textbook improper ferroelectric since the 1960s. Its transition is driven by a zone-boundary mode that carries no dipole at all; the polarisation appears afterwards, quadratically in the primary order parameter, and the crystal is ferroelectric without any mode having been polar.

The hexagonal manganites are the modern ones. There the primary distortion is a tilting and buckling of the oxygen environment — a zone-boundary mode of a non-polar parent — and the polarisation is again a consequence rather than the cause. The parent is hexagonal and the mechanism runs through a threefold axis, which is exactly the feature this essay’s plane enumeration identifies as the only one that supplies improper cases at all.

And the boracites are a third family, notable because their transitions couple polarisation and magnetisation at once, so an improper ferroelectric that is also magnetically ordered gives the magnetoelectric behaviour the magnetic classification permits.

What all three have in common is the zone boundary. A cell-doubling mode carries no dipole by the argument this essay makes, so any polarisation it produces is improper by construction — and the three-dimensional groups have far more zone-boundary wavevectors, with far larger little groups, than the seventeen do.

So the plane’s thin result is not a failure of the enumeration. It is a measurement of how little room two dimensions leave, and it is the same shape of conclusion as several elsewhere in this collection: the arithmetic runs identically in both, and the plane has fewer places for it to land.