[Wien] How to determine the exchange splitting in metals ?
Fecher, Gerhard
fecher at uni-mainz.de
Mon Apr 27 12:13:55 CEST 2020
Stoner exchange (or parameter or exchange integral)
and (magnetic) exchange splitting are two different things.
It seems you did not check the band structures or tried to see what happens when
you are just shifting arbitrarily the density of states.
I strongly suggest that you first read and understand the textbook of J. Kübler
("Theory of itinerant electron magnetism"). It contains very nicely how to
calculate the Stoner parameter or Stoner exchange as well as its q-dependence
and relation to the suszeptibility (see Chapters 4.1 to 4.3 of the 1st edition). He
gives values not only for the exchange splittings (Tables 4.6 and 4.7) but also
for the Stoner parameters (Table 4.2) that are calculated by different methods.
I don't think it is calculated in a straight forward way in Wien2k and you will
have to implement it by yourself. The chapters concerning the Stoner model are
too long to be repeated here in the forum.
Be also sure to check the references in that book.
You also use a very unphysical type of description, there are no shifts to the
left or the right. Energies are either higher, lower, or the same. By the way,
the difference in the number of valence electrons between majority (5.1) and
minority (2.9) is just the magnetic moment in Bohr magnetons, isn't it.
Just some additional remark:
I already explained in detail how to find the energies and splittings at the
Gamma point. If that is not enough for you, then you have to determine the
average energies by yourself.
What you are doing wrong is, you argue with the number of electrons N and not
with their mean energies E^bar, where (as I told partially already before)
(1) N = integral n(E) de
and
(2) E^bar = integral n(E) * E dE,
indeed, with suitable ranges and normalisation. The latter integral you have
to perform by yourself, at least I don't see that it is given somewhere in the
output files of Wien2k.
The band ranges below are given in the case.scf2 files and are for iron approximately
for valence states only:
band ranges from scf2 (energies in Ry)
up occ dn occ at Gamma
bnd 4 0.021-0.365 1.00 -0.005-0.474 1.00 G1+
bnd 5 0.248-0.444 1.00 0.384-0.587 1.00 G5+
bnd 6 0.358-0.612 0.99 0.467-0.772 0.89
bnd 7 0.436-0.613 0.99 0.577-0.773 0.01
bnd 8 0.477-0.634 0.86 0.637-0.797 0.00 G3+
bnd 9 0.522-1.310 0.22 0.721-1.338 0.00
(note the removed degeneracy that I mentioned earlier)
(note also , all numbers in the following are just
approximately (!) given for iron)
If you like to have the mean energies of all valence band, the integrals should
run from 0.021 Ry to 1.31 Ry for majority and from -0.005 to 1.338 for minority
electrons.
However, there are other unoccupied bands that start already at lower energies
and thus overlap with those where one likes to calculate E^bar for,
that is at 1.1 Ry for majority and 1.18 Ry for minority states. (Again you can
use irrep to check the characters of these states.)
Therfore, you may integrate until 6 electrons (1s + 5d) are occupied in both
channels, that is at 1.16 Ry for majority and 1.25 Ry for minority electrons.
Independently, you will have some error if you include the density of states
of additional states (see below remark on Wannier functions).
Principally, you could also calculate the mean energies of the a1g, t2g and eg
states or their sub representations (indeed with similar restrictions) by taking
the band ranges from above, but also with another problem, because Wien2k
calculates their density only within the muffin-tin spheres. One solution might be
to normalize integral (2) by integal (1). Indeed this is the general definition of
the mean value.
(2*) E^bar = integral n(E) * E dE / integral n(E) de.
All these integrals may be easier solved when using Wannier functions to
distinguish between the orbitals and their energy ranges. Further their will be
no restrictions to muffin-tin spheres. I don't know whether something like that
is implemented in the Wannier programs. If not the programming should be
straight forward. Maybe you tell us soon how it works.
By the way: I = Delta/M is only valid if Delta=const does not depend on k.
otherwise you have to use at least the average of I(k).
But overall is seems that I just repeat myself.
PS.: a short version of the relations between band energies, suzeptibility and
Stoner exchange integral I is given in the textbook
Peter Mohn "Magnetism in the Solid State: An Introduction"
that also lists values of I calculated by KKR
for many paramagnetic and ferromagnetic metals with cubic structures.
Fe, Co, and Ni should be in the range of 28-37 mRy
(values from book of Kübler or original work of Janak 1977
values in Mohn are twice as high for unknown reason !!!!).
You may use it to compare with your results.
Ciao
Gerhard
DEEP THOUGHT in D. Adams; Hitchhikers Guide to the Galaxy:
"I think the problem, to be quite honest with you,
is that you have never actually known what the question is."
====================================
Dr. Gerhard H. Fecher
Institut of Inorganic and Analytical Chemistry
Johannes Gutenberg - University
55099 Mainz
and
Max Planck Institute for Chemical Physics of Solids
01187 Dresden
________________________________________
Von: Wien [wien-bounces at zeus.theochem.tuwien.ac.at] im Auftrag von Abderrahmane Reggad [abde.reggad at gmail.com]
Gesendet: Sonntag, 26. April 2020 12:55
An: A Mailing list for WIEN2k users
Betreff: Re: [Wien] How to determine the exchange splitting in metals ?
Thanks Dr Gergard or the full explanation
I want to deterine the exchange splitting which allows me to determine the Stoner parameter value I
If we take the example of Fe, in the paramagnetic state the number of up and the dn electrons equal both 4 and the DOS bands for up and Dn spins are fully symmetrized whereas in the ferromagnetic state a double shift occurs for the Up and Dn bands ( a shift of the Up bands in the left side and the number of the up electrons becomes 5.1 ( the number of the d elctrons is only 4.62 ) and a shift of the Dn bands in the right side and the number of dn electrons becomes 2.90 ( the number of d electrons is only 2.42 ).
I think that the exchange splitting energy as Pr Peter said is the sum of the energy shift of the Dn bands in the right side and the energy shift of the Up bands in the left side.
Is my conclusion correct or not ? and if is it correct how to determine this exchange splitting ?
Best regards
On Sun, 26 Apr 2020 at 09:21, Fecher, Gerhard <fecher at uni-mainz.de<mailto:fecher at uni-mainz.de>> wrote:
Actually it is not clear what you want to do, and for what purpose !
... and answer to "Is correct ?" : NO, the exchange splitting is not the energy difference between two maxima of the density of states.
Check the band structure of Fe
you see that there are in the majority case three bands (states) occupied at the Gamma point,
these are from bottom (lowest energy) a1g (s), t2g (dxy, dxz, dyz) , and eg (dz^2, dx^2-y^2)
in the minority case only the a1g and t2g are at Gamma below the Fermi energy.
(the degeneracy is seen from the dispersion in the Sigma direction)
You should easily find out that there is no unique splitting, but it depends on the state,
and as I already mentioned in earlier mail you have different splittings (including not only exchange splitting) depending on the k-points and their symmetry.
The energies of the bands, or better the ranges of the band energies are (roughly, depending on the k-mesh) given in the case.scf2up/dn files.
To find the correct energies at Gamma you van either have a look in the spaghetti files or better use irrep.
(Note their might be changes in the band series if you have a more complicated system,
and then you have to use irrep do find the correct band character)
The case.outputirruo/dn contains all informations, in a shortened way you will find for iron something like
(note that I cut the values after 3 digits)
for the states at Gamma (using irrep, eigval in Ry, all other information skipped):
from up from down
bnd ndg eigval eigval diff * 13.605 my info: band character
4 1 -0.021 .......... =G1+ 0.005 .......... =G1+ ==> 0.22 eV A1g (s)
5 3 0.436 .......... =G5+ 0.57 .......... =G5+ ==> 1.91 eV T2g (dxy, dxz, dyz)
8 2 0.525 .......... =G3+ 0.728 .......... =G3+ ==> 2.75 eV Eg (dz^2, dx^2-y^2)
you see that there is a very different splitting that depends on the character of the state !
(Principally one could calculate the average (1.9eV all, 2.2eV only d, note the degeneracy !),
but I think that this is an oversimplification and might be used for very rough comparison only.)
To use single maxima of the density of states for determination of the splitting does not make any sense,
just plot the density and shift either up or dn DOS by the value you found to see that it doesn't tell anything.
(The idea of Peter would only work if the up/dn densities are identical with only a rigid shift, what is usually never the case.)
Another way may be to try to find the average energy of the bands, as I mentioned before, but this will only give a very rough idea as it includes not only
the exchange but also the crystal field and possible other splittings. Further one has to take care on suitable energy ranges (see case.scf2) and normalization.
The difference in the band energies for majority and minority electrons, however, is still a good measure for the magnetic energy.
You where also asking for " the exchange splitting between the paramagnetic and ferromagnetic states", it is not clear what you mean ?
Do you mean the shift of the states between non-spinpolarized and spinpolarized up, and between non-spinpolarized and spinpolarized dn ?
Again, there is no unique shift of the states but you may compare the energies at Gamma similar to what I did above.
I suggest to read and understand the textbook of J Kübler "Theory of itinerant electron magnetism", he lists the splittings at different
high symmetry points of the Brillouin zone (for different calculaton schemes compared to experiment) for iron and cobalt.
Ciao
Gerhard
DEEP THOUGHT in D. Adams; Hitchhikers Guide to the Galaxy:
"I think the problem, to be quite honest with you,
is that you have never actually known what the question is."
====================================
Dr. Gerhard H. Fecher
Institut of Inorganic and Analytical Chemistry
Johannes Gutenberg - University
55099 Mainz
and
Max Planck Institute for Chemical Physics of Solids
01187 Dresden
________________________________________
Von: Wien [wien-bounces at zeus.theochem.tuwien.ac.at<mailto:wien-bounces at zeus.theochem.tuwien.ac.at>] im Auftrag von Abderrahmane Reggad [abde.reggad at gmail.com<mailto:abde.reggad at gmail.com>]
Gesendet: Freitag, 24. April 2020 16:42
An: A Mailing list for WIEN2k users
Betreff: Re: [Wien] How to determine the exchange splitting in metals ?
Hello again
I have adopted another procedure as follows:
I have extracted the energy values corresping to the maximum values of DOS in the up and down spin from the files case.dos1evup and case.dos1evdn and I have found the following values:
E (max DOS up = 2.40) = - 0.95185 eV
E (maxDOS dn = 2.80) = + 1.95977 eV
and I have calculated the exchange splitting from this formula dE= + 1.95977 - ( - 0.95185) = 2.91162 eV
Is correct ?
On Fri, 24 Apr 2020 at 14:54, Abderrahmane Reggad <abde.reggad at gmail.com<mailto:abde.reggad at gmail.com><mailto:abde.reggad at gmail.com<mailto:abde.reggad at gmail.com>>> wrote:
Thanks Gerhard for the explanation but I couldn't apply the inforation to get the exchange splitting neither from the DOS nor from the band structure
Now i have some questions about my idea using the DOS picture and I want from both of you to answer me
- is it possible from the files case.outputtup and case.outputtdn to get the energies corresponding to the integrated DOS values for spin up (5.1 e) and spin down (2.9 e) ? in the case o iron Fe
- Is it possible to determine the exchange splitting as follows: dE = E (5.1 e) - E (2.9 e) in abslute value
On Fri, 24 Apr 2020 at 14:52, Abderrahmane Reggad <abde.reggad at gmail.com<mailto:abde.reggad at gmail.com><mailto:abde.reggad at gmail.com<mailto:abde.reggad at gmail.com>>> wrote:
Thanks Gerhard for the explanation but I couldn't apply the inforation to get the exchange splitting neither from the DOS nor from the band structure
Now i have some questions about my idea using the DOS picture and I want from both of you to answer me
- is it possible from the files case.outputtup and case.outputtdn to get the energies corresponding to the integrated DOS values for spin up (5.1 e) and spin down (2.9 e) ? in the case o iron Fe
- Is it possible to determine the exchange splitting as follows: dE = E (5.1 e) - E (2.9 e) in abslute value
I have joinded the case.outputt files or the paraagnetic and ferromagnetic state of iron Fe
Best regards
On Fri, 24 Apr 2020 at 12:26, Fecher, Gerhard <fecher at uni-mainz.de<mailto:fecher at uni-mainz.de><mailto:fecher at uni-mainz.de<mailto:fecher at uni-mainz.de>>> wrote:
maybe use irrep to see how much the bands at the Gamma point are splitted between up and down
(single k-point no shift of BZ before calculating lapw1 -up, -dn; but be carefull which states at Gamma you compare)
Note the splitting depends on k, what you easily see from the bands, therfore a comparison of the bands mmight not be very helpful.
the mean or state resolved splittings may also be calculated by the difference in the band energies for up and down states, that is
integral n(E)up * E dE - integral n(E)down * E dE
where the integrals run over all occupied states of the valence bands, or you use only particular states, e.g. all d pr eg, or t2g only.
(Note the sum is the overall band energy, that you may compare to that of the paramagnetic state, if you wish to do for whatever reason)
Ciao
Gerhard
DEEP THOUGHT in D. Adams; Hitchhikers Guide to the Galaxy:
"I think the problem, to be quite honest with you,
is that you have never actually known what the question is."
====================================
Dr. Gerhard H. Fecher
Institut of Inorganic and Analytical Chemistry
Johannes Gutenberg - University
55099 Mainz
and
Max Planck Institute for Chemical Physics of Solids
01187 Dresden
________________________________________
Von: Wien [wien-bounces at zeus.theochem.tuwien.ac.at<mailto:wien-bounces at zeus.theochem.tuwien.ac.at><mailto:wien-bounces at zeus.theochem.tuwien.ac.at<mailto:wien-bounces at zeus.theochem.tuwien.ac.at>>] im Auftrag von Abderrahmane Reggad [abde.reggad at gmail.com<mailto:abde.reggad at gmail.com><mailto:abde.reggad at gmail.com<mailto:abde.reggad at gmail.com>>]
Gesendet: Freitag, 24. April 2020 11:30
An: A Mailing list for WIEN2k users
Betreff: Re: [Wien] How to determine the exchange splitting in metals ?
Thanks Pr Plaha for the explanation
Now how to determine it through the band structure
Best regards
On Fri, 24 Apr 2020 at 09:14, Peter Blaha <pblaha at theochem.tuwien.ac.at<mailto:pblaha at theochem.tuwien.ac.at><mailto:pblaha at theochem.tuwien.ac.at<mailto:pblaha at theochem.tuwien.ac.at>><mailto:pblaha at theochem.tuwien.ac.at<mailto:pblaha at theochem.tuwien.ac.at><mailto:pblaha at theochem.tuwien.ac.at<mailto:pblaha at theochem.tuwien.ac.at>>>> wrote:
I would do this with the band structure (because this could be
k-dependent), but DOS is also fine.
Just shift up and dn DOS in energy until they overlap as much as
possible. This shift is your exchange splitting.
Am 24.04.2020 um 00:07 schrieb Abderrahmane Reggad:
> Hello wien2k users
>
> I have calculated the DOS of the paramagnetic and ferromagnetic of 3d
> transition metals Ni , Fe and I want to determine the exchange splitting
> between the paramagnetic and ferromagnetic states.
>
> How to do that ?
>
> Best regards
>
> --
> Dr. Abderrahmane Reggad
> Engineering Physics Laboratory
> Faculty of Material Sciences, Ibn Khaldoun University, Tiaret, 14000,
> Algeria
> Tel: +213(0)561861963 - Algeria
>
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