See for example

T.L. Mitran, Adela Nicolaev, G.A. Nemnes, L. Ion, S. Antohe,

"Magnetic behavior and clustering effects in Mn-doped boron nitride sheets",

J. Phys.: Condens. Matter 24, 326003 (2012)

http://iopscience.iop.org/0953-8984/24/32/326003/

Hello,

Sorry, rather a question: What is your opinion, what is the accuracy of the J couplings using i) GGA and ii) GGA+U functional/approaches? I saw today a presentation (on single-molecule magnets) where the speaker reported energies with one digit in cm^-1 (wave numbers). To me it is hard to believe such an accuracy, even tens of wave numbers. In "predictions" I would not expect a "predictive power" of better than hundreds of wave numbers. How pessimistic am I?

Thank you,

apsi

You are trying to map detailed DFT calculations on to a more coarse-grained Hamiltonian, so you would need to fall back on the definition of J. Assuming that you can back out the spins of the atoms in your system and the total energy (E) for different configurations, the Ising model Hamiltonian is given by:

E = -J*[sum over nearest neighbors i & j](s_i*s_j) - mu*H*[sum over atoms i](s_i)

In the simplest case where H (the magnetic field strength) = 0, J is the slope of E vs. the sum of products of s_i for neighboring atoms. s_i is twice the spin state of atom "i" in the z direction (the direction of the magnetic field). This may be an obvious answer, but I'm not sure that there is anything else that you can do to parameterize the Ising Hamiltonian based on your DFT calculations.

I think, If you can find the total energy difference between FM and AFM , then that energy difference (E(AFM)-E(FM)) is equal to N*J*S*S. Where N reperesents no.of nearest neighbors and S is total spin. From this relation may be you can find J value.

You may use the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), Phys. Rev. Lett. 77 3865 (1996) for exchange-correlation energy. It works well in various systems. What is your system?

Thank you all. I used PBE PP for my calculations. I try to get exchange details for YVO3 and YMnO3 in orthorhombic structure.

Wouldn' you say that J rather than being a universal number, it is a conceptional quantity you use when writing down a specific Hamiltonian (Ising or Heisenberg type)? You can use these numbers for different purposes, either to get the low energy magnetic excitations or the much higher energy critical temperatures. My understanding is that the latter is much harder to predict...even if you have a J (I may be wrong, though).

Concerning the first alternatibve (effective J('s) for low energy magnetic Hamiltonians) there is a technique frequently called "downfolding", to achieve this, starting from DFT calculations. One of the developing groups was that of O. Anderson (at the solid state physics Max Planck Institute in Stuttgart, Germany). What I got to see from their work didn't look like a trivial thing to me. But I have seen impressive results. So, looking at their publications may help.

Thank you very much K. Fauth. What I mean is J only as you said and I want to calculate as exchange energies (J) for different spin arrangements (G, A, C or F) in YVO3 and LaVO3 to come to the conclusion that which of the configuration is more favorable compared to other. Energy difference in these arrangements but my quarry are small how to get these energy by DFT (Quantum espresso or ABINIT) in future with some external fields. Please help me to get some insight to this.

In that respect, what I have seen in the past is that different spin structures were forced upon the system (i.e. paramagnetic, ferromagnetic and/or antiferromagnetic or whatever other ordering type). Then you compare the total energies of the different computations and decide on the minimum energy configuration within your theoretical description. If those differences are small, then you will have difficulties in deciding the matter from DFT alone - that would be my guess. And it might be that the ground state is decided by terms which are hard to account for within DFT (e.g. Coulomb correlations).

I won't be able to comment beyond that, unfortunately, and others may have more valuable recommendations for you.

Thank you very much Fauth. I agree that usually magnetic structures are forced that making me difficult to get angle between spins in canted antiferromagnetic type structures.

I don't know whether QE provides an opportunity to calculate canted spin structures. I reckon you would have to formulate the adequate set of basis states first (with canted spins) and then caulculate the total energy as a function of canting angle? Doesn't sound like really simple but not like undoable neither. But I don't know whether it's feasible without extra programming. Maybe there have been developments in other packages?

Thank you so much. QE can do canted spin structure and I did work for LaCrO3. But I got the problem of band gap which is small compared to experimental one. in QE along with non-colliear arrangement one cant use energy U in the calculation. Please see our work. https://www.researchgate.net/publication/236163872_Ground_State_Electronic_and_Magnetic_Properties_of_LaCrO3_System.

I want to improve on it.

Thank you so much for your help. Now I can be able to tell the exchange energy (J). If you come across some ref related to this please let me know.

Maybe there's a way to mimic U by orbital polarization or SIC or some other scheme which might continue to work with noncollinear spin states?

(1) the simplest way is to indeed to extract J from the total energy difference for different magnetic configurations, but one should remember the Heisenberg model is valid only if the orbital configuration is fixed rigidly by the crystal field. Strictly speaking, it is not necessary true and by constructing the Heisenberg model for the oxides you can miss several important effects, related to the activity of the orbital degrees of freedom; (2) Downfolding: if you can construct the Wannier functions, you can calculate the transfer integrals and then evaluate superexchange interactions (assuming some values of U or by calculating it); (3) If you can do non-collinear, one more possibility is the frozen spin spiral: calculate total energy (one-electron part is typically sufficient) for different spin-spiral vectors q the cone of the spin-spiral theta, then take second derivative wrt theta and take the Fourier transform. This will give you all possible J's. Some works on YVO3, LaVO3 and YMnO3: PHYSICAL REVIEW B 74, 054412 (2006) and PHYSICAL REVIEW B 86, 144406 (2012).

Thank you so much @Solovyev. I start working to my problem based on this paper only to get correct J value. I am new to this field. First of all understanding what computer (code) is doing is big problem for me. Roughly I get some idea what its doing but was unable to convince myself whether the total energy difference for different spin configurations may give correct J value as I could not include in the calculation(Quantum Espresso) spin-orbit coupling along with U. If I not understand wrongly the inclusion of U going to add to orbitals symmetrically or equally to all 3d orbitals (Please correct me if I am wrong!). In particular, Non-collinear spin arrangement for LaCrO3 (I think does not have orbital degree of freedom in its ground state!) I did the pDOS (DOS) calculation invoking spin orbit coupling (our work http://www.scientific.net/AMR.585.274), indicate Mott-Hubbard type insulating (not of charge transfer type Ref: T. Arima, Y. Tokura and J. B.Torrance Phys. Rev. B 48, 17006–17009 (1993)) behavior along with observed weak ferromagnetic nature (by magnetization measurements our experimental work still to publish).I am very much thankful to you. Please help me to understand third point of your discussion. I am so much interested physics related this type of materials though my PhD work have been on magnetic and optical properties of these materials experimentally.

If you need only isotropic J, you can forget about spin-orbit coupling. The "frozen spin spiral" is a kind of perturbation theory in terms of small rotations of spins near some equilibrium magnetic state. The idea is slightly rotate the spins (or do some small canting from the z-axis) and then modulate the obtained structure as the spin spiral (employing the generalized Bloch theorem, where each translation is combined with the rotation of spin). The second derivative with respect to the canting will give you J in the reciprocal space (for each spin-spiral vector q). It will give you some "local" Heisenberg model, which is valid only near this equilibrium state. I would like to emphasize that the Heisenberg model can be always defined locally (i.e. near some equilibrium) - this is a general property of the second order of perturbation theory, which can be always presented as a sum of pair interactions. This model will tell you about the type of your magnetic equilibrium: whether it is minimum, maximum, or a saddle point. Whether you can use for the analysis of the transition temperature is a big question (and I would say, a fundamental question about general validity of the Heisenberg model). In order to test it, you can compare with J extracted from total energies for different magnetic configurations (say F, A, C, and G, as it was already suggested). If two sets of parameters J will coincide, then your Heisenberg model is well defined and you can for all kind of purposes (local stability AND the transition temperature). However, such situations are rare (e.g., only if the dominant mechanism is superexchange and the orbital configuration is frozen by the strong crystal field). For your materials, the Heisenberg model is only a approximation, but because of different reasons. The best example is YVO3 (superexchange works and orbital configuration is reasonably frozen). LaVO3 is already worse (superexchange is applicable, but the orbital configuration is not totally frozen by the crystal field). In YMnO3 the orbital configuration is frozen by the huge crystal field, but ... it is necessary to go beyond superexchange (i.e., beyond the 1/U expansion), otherwise you will not be able to obtain the longer-range magnetic interactions, which are important for the magnetic inversion symmetry breaking and multiferroic properties of YMnO3.

In quantum espresso, the exchange energy is found in the last iteration of self consistence field calculations..

Dear Gboyega, are you sure we are talking (writing) about the same thing here? Consider a ferromagnetic material. It has interatomic exchange coupling. If we (try to) represent this by a Heisenberg type Hamiltonian, then what is the appropriate exchange constant which governs magnetic order? I doubt that this information can be found in the way you suggest.

Edit: If you know or believe otherwise, please share your experience and/or be more explicit and informative about the procedure(s) you have in mind.

Dear Kai,

The quantum espresso developers have a very robust interactive forum for users/developers where answers to several encountered problems are already provided.

From a discussion with D.Ceresoli, I understand that it is not easy to obtain J in many calculations; however, with QE, there could be two ways to it: First, determine the spin resolved density of state projected on magnetic centres- the splitting between the (peaks) spin up and spin down DOS should give an estimate of J. Alternatively, J can be determined from the diference (between the anti-ferromagnetic and ferromagnetic states) divided by spin magnitude. If you use QE, then you have to set correctly the flag starting_magnetization(..). There could be many route to determination of J, however, since Brajesh uses QE, the above methods may serve his purpose.

Thank you.

It is dangerous to equate the band splitting with the Heisenberg exchange energy, particularly so in metals. Band splittings give way larger values, very much incompatible with the observed Curie temperatures.

The second answer you now provided comes down to comparing the relative ground state energies of different magnetic configurations. This was already discussed above (check popular answers, for example, including limitations formulated there).

Such a procedure is, however, different from looking into "the last iteration of self consistence calculations", as you wrote before (unless, of course I am wrong here. I have no hands-on experience with QE). This is the point I wanted to make in the first place when I challenged your first answer. It looked all too simple to me.

(Again, I may be wrong. And I'll be ready to learn :-)

Dear Kai,

What is listed in the last iteration of the self consistence field calculation is the exchange energy contribution to the total energy. However, I thought my last contribution addressed ways in which the Exchange J can be calculated using QE. To make my points clearer, one has to perform DFT+U (Hubbard correction) noncollinear magnetization and spin-orbit interactions using relativistic pseudopotentials (this is well explained by the developers in QE manual). I am quite aware of the effects of large band splitting metals; in the case of "ceramic LaCrO3", which Brajesh is considering, I believe QE will give good approximation of J.

Thank you.

Thank you Adebayo and Kai. Calculations for LaCrO3 done exactaly as Adebayo sayed. But could not get J values in end of QE.

Maybe attached paper can be helpfull.

Article Electronic and magnetic properties of the monolayer RuCl$_3$...

I think using TB2J with QE DFT calculations will provide a good estimate of J values.

https://tb2j.readthedocs.io/en/latest/

Article TB2J: A python package for computing magnetic interaction parameters

https://github.com/mailhexu/TB2J_examples/tree/master/Wannier

These might help you.