You should look at the problem in the following way perhaps: The antisymmetric D-M interaction is the relativistic correction to the usual superexchange interaction and its strength is is proportional to the spin-orbit coupling. This interaction favors/produces non-collinear spin ordering, viz. spiral spin ordering. Once it produces the spiral state the lattice reacts (relaxes) to it by the magnetoelastic effect or exchange striction (minimizes energy) and that cases the atomic shifts or ferroelectric polarization. So the inverse effect is nothing but the the efforts of the lattice degrees of freedom to repair the damage done by the D-M interaction. One should not perhaps call inverse D-M interaction but the lattice response to the same D-M interaction which is responsible (the culprit!). I am deliberately making is simple because theoreticians sometimes make it unnecessarily complex (deliberately?) and I am sure they will not like this answer.

You should look at the problem in the following way perhaps: The antisymmetric D-M interaction is the relativistic correction to the usual superexchange interaction and its strength is is proportional to the spin-orbit coupling. This interaction favors/produces non-collinear spin ordering, viz. spiral spin ordering. Once it produces the spiral state the lattice reacts (relaxes) to it by the magnetoelastic effect or exchange striction (minimizes energy) and that cases the atomic shifts or ferroelectric polarization. So the inverse effect is nothing but the the efforts of the lattice degrees of freedom to repair the damage done by the D-M interaction. One should not perhaps call inverse D-M interaction but the lattice response to the same D-M interaction which is responsible (the culprit!). I am deliberately making is simple because theoreticians sometimes make it unnecessarily complex (deliberately?) and I am sure they will not like this answer.

I must add that the atomic shifts or electric polarization is caused by the D-M interaction (main culprit) through exchange striction. So both these effects have to be considered together. But it is the same D-M interaction whose strength is propprtional to the spin-orbit coupling constant.

Thank you!I like this answer, I can get what is the physical picture in the lattice!

It is nice to guess what actually causes atomic shifts in multiferroics but to be frank the theoreticians do not know really. The debate whether D-M interaction is important or not in the multiferroic RMn2O5 is still not settled. The most important thing for the experimentalists should be to determine these atomic shifts in multiferroic materials. But these shifts are very small and extremely accurate diffraction measurements are necessary. I have been trying to measure these shifts in hexagonal manganites (where D-M interaction is not effective) and also in another class of multiferroics RMn2O5 9R = rare earth element) but so far our success is very limited. Only recently our recent polarized neutron diffraction (flipping ratio) had some success.

Which multiferroic materials are you looking at and what are the techniques are you employing?

I recall a study of Radaelli and Chapon about the possible origins of Ferroelectricity in RMn2O5..We are dealing with bulk delaffosites (AMn+3O2) where FE is induced by ICM order.So far we have employed CW and TOF NPD, HRTEM, INS. Analysis of the NPD is not complete yet, but I suspect that DM or inverse DM are likely to occur.I couldn't see the difference in the hamiltonians of the DM and the inverse DM, so it is really good to know the actual impact -difference in the lattice..

I will probably complicate the matter but I’d like to comment a bit. I also hope that you will excuse some self-citation. You might find interesting the paper J. Phys.: Condens. Matter 24, 266002 (2012), where we show that things can be simpler in the microscopic origin of the magnetoelectric effect. Specifically we show that the magnetoelectric effect can be due to noncentrosymmetry of local surroundings (crystal field) of ions. We find that the spin-current model is not relevant to magnetoelectricity in MnWO4, whereas the inverse DM model does not reproduce all of the possible interactions allowed by the macroscopic symmetry. Furthermore we actually find that in the original paper on inverse DM this effect cancels out in TbMnO3 when one considers the macroscopic orthorhombic symmetry. Therefore, one should always consider the macroscopic lattice symmetry when interpreting the magnetoelectric effect using one of the models, which is arguably not the case in most of the papers on magnetoelectric properties of crystals.

Also, since you are dealing with delafossites you might find interesting the paper in Ferroelectrics 438, 101 (2012). I can provide pdfs if you like.

Concerning the DM interaction I can add that recently I’ve seen an hour-long presentation on the matter and the author concluded with the following problems to be solved: location of the Dzyaloshinskii vector, sense of the Dzyaloshinskii vector, and oxygen contribution to the Dzyaloshinskii vector. Therefore, despite a long history of DM interaction there are still important unresolved problems.

Thanks for recommending the papers, I will read them and get back to your comments for better understanding. I definately agree that the macroscopic lattice symmetry plays the crucial role..My question is more general, in the sense of what is the difference between the DM interaction and the inverse DM interaction. In a simple way(if you can simplify thsi) it looks like a combination of spin orbit and superexchange interaction that causes the shift of the non magnetic anions (O) in both cases, so their difference is not that clear as well as the difference in the direction of the induced polarization in magnetoelectrics..

That’s right, if you consider the spin-orbit coupling and super exchange, then you arrive at D[Si x S(i+1)].

Here D is proportional to spin-orbit coupling and can in some cases be associated with oxygen displacements. In simple words the difference between DM and inverse DM is the following. If you have nonzero D, then this term in the Hamiltonian results in spin canting (e.g. weak ferromagnetism). This is DM interaction.

On the other hand if you have canted spins (like what can happen in spirals, for example), then this term may induce nonzero D, which may result in ion displacement. This is inverse DM interaction.

However, I think that the notion that this term is responsible for both the spiral and the emerging polarization (i.e. D is nonzero in the paramagnetic phase and changes in the magnetic phase in such a way that polarization arises) is wrong.

This a nice and really elegant way of describing a phenomenon which in most of papers appears somehow vague.Thank you.As for the origin of polarization,I don't have the theoritical experience to give a full explanation, so the way I get it is that when D is non zero there is the breaking of the inversion symmetry, necessary for a polar group, and thus we can have polarization. In that way DM(proper or inverse) is highly related to the appearance of ferroelectricity in well studied magnetoelectrics such as TbMnO3, Ni3V2O8 etc..

No, D is a microscopic thing which describes two interacting spins. It is not directly related to inversion symmetry breaking. In centrosymmetric crystals D can be nonzero for some interacting pairs of spins like for example in alpha-Fe2O3. This results in antisymmetric exchange.

Hi,

I am very sorry to say that now I am again confused a bit. What does give the polarization , the DMI or IDMI. I get the way so far by your description that DM induce spin canting (as a result weak ferromagnetism but no polarization) and IDM is lattice response of it which gives rise to polarization. And one more think when there is a magneto electric coupling which one should we think first the DM or IDM. And is there any competition between DM and IDM?

Hi Chandan, I am afraid I agree with you.Now, I understand the difference of the DMI and DM, but I am still looking for the difference-relation to the induced polarization and its direction when it comes to magnetoelectrics..

Well, first of all there is only DM interaction and no such separate thing as inverse DM interaction. The DM interaction is given by D[S1 x S2]. D is a microscopic characteristic of two interacting spins. It depends on orbital moments and is proportional to spin-orbit coupling. This term in the Hamiltonian can be responsible for weak ferromagnetism for example. E.g. in alpha-Fe2O3 there occurs an antiferromagnetic phase transition, upon which a weak ferromagnetic moment arises. This is due to the fact that this term favors canting of spins, which results in noncollinearity and weak ferromagnetic moment. Similarly if alpha-Fe2O3 were ferromagnetic, this term would have resulted in weak ANTIferromagnetism. Thus, D is nonzero and does not result in electric polarization.

Magnetic ordering can be complex. The magnetic cell, for example, can be a multiple of the crystallographic one. Moreover a spatial modulation of the magnetic structure can sometimes arise. This results in complex ordering such as spirals, cycloids, etc. In such structures spins can rotate by small angle from cell to cell. Therefore, this term is often thought of as the source of the complex spatial modulation. However, one can obtain incommensurate structures without this term by competing usual (super)exchange (e.g., ANNNI model).

There is no such separate thing as inverse DM interaction. The term “inverse” is used here to point that there is some complex magnetic structure (e.g. a spiral) with noncollinear spins (such that [S1 x S2] is nonzero) and the DM interaction produces nonzero D or some increment to D. This in some cases may lead to macroscopic electric polarization.

Therefore, the logic is that DM interaction with nonzero D produces noncollinear spins. Whereas inverse DM is that noncollinear spins produce nonzero D or an increment to D, but through the very same DM interaction.

As I noted above, I think that this term cannot be responsible for both the spiral and the electric polarization.

As to how to determine the direction of polarization that can be induced by inverse DM interaction, it is a question of the magnetic structure and crystal symmetry. First you should consider the interacting M1-O-M2 clusters and their symmetry. Have a look at the original paper by Moriya, there is a discussion on the symmetry of the cluster and properties of D. However, it is not clear how D is related to atomic displacements. You could look for it in textbooks, but I haven’t found any good one on the topic. Then you will know how the atoms displace in each cluster. Finally you should consider how these clusters are packed into the crystal lattice, because the packing may result in zero macroscopic polarization. This is what happens, from my point of view, in the original paper on inverse DM interaction. In most of the papers one considers the general consequences of the spin-current or the DM model about the direction of polarization for certain spin directions in a cluster, but forgets about the crystal structure and how these clusters are packed into the structure.

Finally, I have to add that it is an experimental fact that electric polarization most often arises in modulated (incommensurate) magnetic structures. This has led to pervasive thinking that it is obligatory to have some kind of a spiral in order for polarization to arise. Avoid this thinking, because polarization can arise in commensurate magnetic structures.

I am not able understand how spin-orbit coupling lead to displacement of atoms and how does then D then relate to inversion symmetry breaking?

Spin-orbit coupling is one of the strongest interactions of the spin subsystem with the lattice.

You probably can think about it in the following way. The electron spin interacts with the magnetic orbital moment of the electron.

Thus, if the spin direction changes so does the orbital moment. The orbital moment is determined by the electron orbital (a cloud).

So the spin infuences the electron cloud (and viceversa), i.e. influences the electron charge distribution. If the spin changes the cloud

configuration, then the lattice might want to respond to it as well to decrease electrocstatic energy for example.

Such a response of the lattice can consist in polar distortions. This will result in electric polarization.

However, in our paper we showed that one can consider only the electron orbitals in order to describe the magnetoelectric effect.

Please look at this paper

Spin-spirals and inverse DM are two the most misleading suggestions in the field of multiferroics. In many popular systems, the magnetic ground state is not the spin spiral and the ferroelectric activity is cause by other mechanisms: for instance, the deviation from the perfect spin-spiral alignment, rather than the spin-spiral itself. For example, using this picture (of deformed spin spiral magnetic structure) one can understand the behavior of orthorhombic manganites (like TbMnO3) and MnWO4. It is not at all necessary to appeal to the spin-spiral ground state and the IDM mechanism.

Book "Magnetism III" edited by Rado, Chapter 4 By Bertaut can give a clear picture about the direction to get find out about D vector. Another paper by Goshen et. al PRB (1970) 2, 4679. which suggest doubling of unit cell in antiferromagnetic system may lead to ferroelectricity.

I have a question: How vector D is related to ferroelectric polarization P? Though I have seen some theoretical paper but I could not convince myself to a logical conclusion.

Thank you all for the suggestions and the contribution in this question.

What I found on the subject, in short:

from Kimura on the "Magnetoelectric Hexaferrites"Annu.Rev.Conden.Matt.Physics, 2012, 3, 93-110) based on Jia's work (Chenglong Jia, Phys.Rev.B 74, 224444 ) he gives the equation which shows that the induced polarization is depending on three terms, one of which is the DM interaction The other two mechanisms that can contribute in the induced polarization is the exchange -striction and the d-p hybridization . The inverse DM or the spin current model, first proposed by Katsura et al ( Hosho Katsura, PRL 95, 057205 (2005)) explains the induced polarization in cycloid magnets (such as TbMnO3).

However, in a proper screw axis magnet, like the CuFeO2 and other delafossites, the eij ˟(Si ˟Sj) of the DM interaction is on average zero, so the DM can not explain the induced polarization. Instead the two cluster model proposed by Jia (same PRB reference of above) and the spin orbit interaction, which are discussed analytically by Taka-hisa Arima, (J. Phys. Soc. Jpn,Vol. 76, No. 7, July, 2007, 073702) and later on also by Seki (Adv. Mater. 2010, 22, 1554–1565) seems to hold for the polarization of the proper spin screws like the delafossites CuFeO2, CuCrO2.

Some comments (just in order to promote the discussion!). (1) Each mechanism should be `material specific'. At least it should depend on the proper symmetry of the material and the way how this symmetry is lowered by the magnetic order (which is not necessary cycloidal. Actually, why should it be "cycloidal" - the are many other possibilities!). (2) In order to lower/break the symmetry by the magnetic order, it is essential to have competing magnetic interactions: of the nonrelativistic origin (Heisenberg), relativistic origin (DM and/or anisotropy) or both of them. But, again, there is no universal scenario of the magnetic inversion symmetry breaking, such as "spin-spiral" or "cycloid" order. It depends on the actual symmetry of the material, and can be different in each case. I would say that the most difficult part of the theoretical analysis of multiferroics is to find the true magnetic ground state! (3) Is rather meaningless to discuss the FE polarization (or try to write some general formula for the FE polarization) without addressing the problem of the true magnetic ground. If the same arguments are applied to the true magnetic ground state, the story can be completely different. Some example (note that in the first two publications some scaling factor for the theoretical FE polarization was missing - it was corrected in subsequent papers): Phys. Rev B 82, 094425 (2010) [for BiMnO3], Phys. Rev. B 83, 054404 (2011) [for TbMnO3, note that it is not necessary "cycloid"], Phys. Rev. B 87, 144403 (2013) [for MnWO4], Phys. Rev. B 86, 144406 (2012) [for HoMno3 and YMnO3, note that the origin of the E-phase can

be understood as a competition of isotropic and anisotropic interactions, without exchange striction], Phys. Rev. B 87, 144424 (2013) [model for orthorhombic manganites, explaining results of first principles calculations]. Sorry for this

"self-propaganda"!