JEMS2020 — Orbital ordering and magnetoelectric coupling in FeCr2O4

В декабре 2020 года я выступал с устным докладом на большом симпозиуме JEMS2020. Удалось разгадать механизмы взаимодействия спинов 3d электронов в орбитально-вырожденном состоянии с электрическим полем. Задача очень актуальная, так как класс соединений с такой связью — мультиферроики — одни из самых необычных современных мультифункциональных материалах с широчайшей нишей потенциальных практических применений.

Ниже приведены слайды с моей речью и некоторыми дополнениями про установление орбитального порядка в упомянутом соединении.

There is no doubts, nowadays, multiferroics as a novel calss of next generation material draws the heighterned attention due the wide variaty of possible applications, especially multiferroics with a strong magnetoelectric copuling. They exibit both porpeties of ferroelectric and ferromagnet. Multiferroics with a strong coupling between spin and electric field are the most interesting. draws the heightened attention due to the wide variaty of possible applications, for instance, as a memory device. It the basic scenario we can control the magnetization applying the external electric field and vice versa (or it is so-called voltage controlled ferromagnetism).

It it important to note that our mechanism is connected with orbital ordering phenomenon, and it is also needed to be claryfied.

On the slide we listed the recent reviews of proposed mechanism of magnetoelectric couping. This is still highly develoing area since the microscopic theory is not entire clear yet. There are still a lot of questions to the origins The microsopic theory of mgnetoelectric coupling is not entirely clear yet, and is still in development. Here professor Mikhail Eremin and I  suggest new possible microscopic mechanism involving combined action of the odd crystal field, spin-orbital and exchange coupling in the orbitally ordered state of \bf{Fe Cr_2 O_4}. We also claryfied the microscopic mechanism of orbital ordering.

It it important to note that our mechanism is connected with orbital ordering phenomenon, and it is also needed to be claryfied.

This \bf{Fe Cr_2 O_4} compound is worth to be studied due to the discovery of the large electric polarisation in ferrimagnet phase which coexists with orbital ordering of iron ions.

At the room temperature it is a regular cubic spinel FD-255, where thetrahedral and octahedral sites occupied by iron and cromomium, respectively.

As you can see the polarization almost saturates when the electric field is applied near Curi temperature. As it was mentiooned by the authors  there is an evidence that polarization is connected with Jahn-Teller ions the iron Fe here. By the looking on the inset inside the plot you can find Cobalt compund, which has no orbital degeneracy and no Jahn-Tller effect, has a rather weaker polarization. But!

The next thing I woud like to notice. Comparing it to the sulfide, where the situation is a bit different, the magnetic order appers firstly and then an orbital ordeing with spontangerous polarization occur.   We can conclude that the polarization exists only when both spins and angular mones are ordered .

In the persence of magnetic field, as it was expected, these is some changes of polarization magnitude. They are relatively small due to the averaging by eachi crystalline in a powder and perhabs, there is a high magnetic anisotropy field in this compound.

To invesigate the problem, firstly we did have to model energy levels of \bf{Fe^{2+}}.

We used the superpositions model where the total energy is a sum of energy of each pair metal-ligand. There are four oxygens located at the vertexes of the terahedron.

The energy of the each pair is characterized by the intrinct parameter \bf{a^{(k)}(R)}.

To estimate them we applied Exchange Charge Model commonly used in opticical spectroscopy, as the advantage it has only one tuning parameter, determined from the experiment. From advantages of this model it takes into account covalency effects and overlapping besides elecrostatic interaction.

The novelty of our approach is electrostatic d-d correction, firstly mentioned by Kleiner.

Basically, if we expand the electrostatic potential of the ligand like I showed on the slide. The first term stands for the point change approximation commonly used in most calculations, which is rough. But the second one approximates the effect of spreading of change density, and makes the electrostatis interaction more realistic. It means that chagre density penetrates to the shell of the iron. Presented form turned out to be more convinient to integrate than any other expansion.

Our estimation are shown on the slide for the components of crystal field, as you can see this correction does not do much when k equals 3 and 4 which stands for cubic symmetry. But it has an impact on \bf{a^{(2)}} wchich enters into linear Jahn-Teller constant copuling electrons with modes of the lattice.

One more thing is that the tetrahedron has no inversion center, that is the reason why there is very high odd crystal field — component \bf{B^{(3)}_{2}}. It will play a crusial role in magnetoelectric coupling.

The lowest state is orbital doublet. Lets move on to the next part.

The coumpund undergoes orbital ordering phase transition with reducing the symmetry to tetragonal. According to the references cited on the slide (it basically the diffraction data) every tetrahedron undergoes a compressions along the c-axis keeping the same volume. And there are two types of tetrahedral fragments rotated around the c axis by 90 degrees with respect to each other.

It can be expected, because of Jahn-Teller effect the orbital degenerated level is unstable and system will reduce the symmetri in order to decrease the total energy.

If we consider the adiatic approximation, isolating each tetrahedron that adiabatic potential for orbital doublet will has multiple minima.

If we minimize the total energy of the crystal including interaction with deformation and the energy stored in the lattice (I mean the strain energy) there will be multiple or infinite minimums with respect to the order parameters. In this case the role of latter plays deformation tensor or lattice constants or generalised coordinates of the tetrahedrons.

To eliminate the disorder we need some corellation between ions to bring them into the same state. Here we found possible two channels whoose energy is sufficient to compete with termal fluctiations.   

The first one corresponds to electric quadrupole-quadrupole interactions of 3d electrons on the nearest sites. The second one the efeective interaction via ecxchagine of acoustic phonons.

It looks like ions play a snowball or throw boomerang.

Dmitrii Tayurskiy. The head of KFU, Institute of Physics, Kazan

This was developed recently by us 1 year ago  for a cubic medium. This operator depends on intrinc parameters of crystal field operator. Using the cricital temperature as a criteria for checking  the type of ordering between two sublattice we found ferro-type is the most favourable. As it was indicaded in the experiments.

We used the mean field approximation method.  By minimizing the free energy we obtained the following values. As you can see, it is pretty close to observed one. And if we neclect the d-d correctin it will jump pretty high. It proofs that our imporvemnts works well.

The minima was in a favour of ferro-type ordering

As I mentioned there are two sublaatice: Fe1 and Fe2. We tried to distinguish them while evaluating the hamiltonian, but the critical temperature was in in a favour of clear ferro-type ordering. It agrees to the experiment. And as you can see, the impact of density-density correction on the critical temperature is quite significant.

It is worth to note  effective coupling via phonon field and quadrupole interaction have the same order of magnitude after summing over the lattice.

So below T_{OO} our orbital doublet splits and the lowest becomes theta  Mossbauer and diffraction data confirm it. Below the critical temperature the doublet splits and the lowes one becomes theta state. Mossbauer data confirms it.

That is all we need to start our envestigations of magnetoelectric coupling.

There is one more problem. The electric dipole operator is nonzero only between the the states with opposite parity.

As It was propesed in works of Judd and Ofelt.

We take into account mixing of configurations with opposite parity.

As it was suggested by Judd and Ofelt, the odd crystal field on tetrhedral site admix the exited configuration with opposite parity to the ground one. In modern notation it can be written as efective operator \bf{H_E} acting inside the pure ground configuration.

The major hassle is to find these parameters small \bf{d^{(1k)p}}.

Basically There are two contributions: from the crystal field operator and covalency bond. We used thechnic mentioned in the reference [3] to esimate them.

Now lets see how spins are couplied to the electric field!

In the orbitally ordered state our ground state is pure 5theta multiplet. Combining spin orbit cpoupling with HE operator in the third order of pertuberation theory we derived the frirst mechanism coupling spin with the external electric field. The final expression is at the bottom of the slide. I would call it single-ions mechanism.

This is written as a frist term of equation at the bottom of the slide.  By the looking at it is easy to understand that elecctric polarization goes zero when the magnetization lies along the any froufold symmetry axis.

The second mechanism has a form of Dzyaloshinskii-Moria mechanism, it demands the noncollinearity between Fe and Cr spins. It also was derived in a simillar way to the frist one, but instead of using spin orit cpiling twise, we alternated with exchange interaction in the exited state, nor the ground. The the most effective path we found is illustraited on the picture via brinding oxygen ion.

To estimate the value of plarization we need to known the detailed spin structure. Since neutron scattering had not been measured yet, we did have to made several assumptions.

  1. The direction and the magnitude of iron spin is propotional to hyperfine field mesured on the nucleus \bf{{}^{51} Fe}.
  2. Monodomain case.
  3. The angle between Fe and Cr is the same as for related compound with vanadium.

As you can see in the table our values agrees with measured ones by the order of the magnitude. And as mestioned by authors the total value can be reuced significantly due to averaging over all orientations

At the temperatures above 40K spin tends to rotate towards a-axis, thus the first single-ion mechanism gets weaker, while the second one dominates.

The polarization above Curie temperature in the experiment exists. There is a possible explanation that it caused by singl-ion mechanism due to a short range magnetic ordering.

And as mestioned by authors the total value can be reuced significantly due to averaging over all orientations since they did not have a signlecrystal.

A few notes regarding geometric frustration

The existance of two sublattice of iron ions I mean Fe1 and Fe2 was a big problem. Because, the C and B parameters changes have on opposite sing in the front for Fe2 sublattice.

When I tried to replicate the collinear spin structure, after averaging over all ions I got zero polarization for the frist mechanism. But I still has the second one, I canted Fe and Cr spins like in spin-ice structure, so it also leaded to zero.

I think It definitly means that the spin structurre even in the ferrimagnet phase seems to be more complex than typical collinear ferrimagnet.

I hope our invesigation in theory will stumulate other researches to develop this topic and perform a neutroin scattering to get detailed data about spins.

Thank you for your attention.

Have a nice everning. Bye.

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