The calculation of the magnetic moment from the DOS picture ( Integrated DOS )

https://www.mail-archive.com/wien@zeus.theochem.tuwien.ac.at/msg19926.html

We will calculate the magnetic moment of 2 ferromagnetic metals governed by the Stoner model within the itinerant magnetism using the DOS picture from ths spin polarized calculation.

Example 01: α-Fe which crystalizes in the bcc structure

We will take the example of the α-Fe which crystalizes in the bcc structure. Using the traditional spin polarized calculation we have found the value of magnetic moment of the Fe atom as follows:

--- MMT -----------
in  1 files:
Fe-bcc-fm.scf::MMTOT:  SPIN MAGNETIC MOMENT IN CELL     =    2.17273

Now using the DOS calculation, we find the values of the occupation numbers of the up and dn electrons which allows us to calculate the magnetic moment of the Fe atom as follows:

From the file case.outputtup we find the occupation of up electrons

EMIN=  -0.06788 EMAX=   1.10008 EFACTR=    499.99996948 ESTEP =   0.00200

 ENERGY BAND    1 THROUGH   12 ENERGY CHANNEL:    1   TO  584
 NUMBER OF K-POINTS:         165
 NUMBER OF TETRAHEDRONS:        5734
 dostot in rxesw  0.0000000E+00
#  BAND   12         
#EF=   0.60008     NDOS= 7     NENRG=  584    Gaussian bradening: 0.00300
 NUMBER OF ELECTRONS UP TO EF         :    5.0941

From the file case.outputtdn we find the occupation of dn electrons

  EMIN=  -0.06788 EMAX=   1.10008 EFACTR=    499.99996948 ESTEP =   0.00200

 ENERGY BAND    1 THROUGH   12 ENERGY CHANNEL:    1   TO  584
 NUMBER OF K-POINTS:         165
 NUMBER OF TETRAHEDRONS:        5734
 dostot in rxesw  0.0000000E+00
#  BAND   12        
#EF=   0.60008     NDOS= 7     NENRG=  584    Gaussian bradening: 0.00300
 NUMBER OF ELECTRONS UP TO EF         :    2.9064

Both nombers form the integrated DOS

Now we can calculate the magnetic moment as follows;

M = Nup - Ndn = 5.0941 - 2.9064 = 2.1877   

We can plot the integrated DOS ( total number of electrons ) using origin tool as in the following picture taken from the book " Constitution and Magnetism of Iron and its Alloys " 

At Ef, the integrated DOS, shown with the dotted curve in Fig. 2.1 0 a, gives the number of electrons in the d-band. For iron this corresponds to Nup + Ndn =8. An occupation at Ef of Nup =5.1 for the Up-band (majority band) and Ndn =2.9 for the Dn -band (minority band)(Fig.2.lOb) gives (Nup - Ndn )=2.2  ${\displaystyle \mu _{\mathrm {B} }}$ for the magnetic moment, in good agreement with the experimentally determined value 2.22  ${\displaystyle \mu _{\mathrm {B} }}$.

In α-Fe, EF lies typically in a minimum of the minority band DOS, and the majority band is not completely filled. The Fermi energy cuts the majority DOS curve at the upper shoulder, so that about 0.2 to 0.3 d-states are unoccupied. For a full majority band, one would expect a magnetic moment of about 2.4 to 2.5  Because of this feature in the DOS, a-Fe is referred to as a "weak:" ferromagnet, as opposed to the "strong" ferromagnets Co and Ni (and y -Fe in the ferromagnetic high-spin state; cf. Fig. 2.20), all with completely filled majority bands (see the Slater-Pauling curve shown in Fig. 4.3).

Reference:

https://materials-mag-algerien970.blogspot.com/2020/04/constitution-and-magnetism-of-iron-and.html

Example 02: Ni which crystalizes in the fcc structure

Calculated magnetic moment using the traditional spin polarized scf calculation

Ni-fcc-DOS.scf::MMI001: MAGNETIC MOMENT IN SPHERE   1    =    0.63049

Calculated magnetic moment using the  spin polarized DOS calculation

the case.outputtup file

 EMIN=  -0.19702 EMAX=   1.02446 EFACTR=    499.99996948 ESTEP =   0.00200

 ENERGY BAND    1 THROUGH   13 ENERGY CHANNEL:    1   TO  611
 NUMBER OF K-POINTS:         165
 NUMBER OF TETRAHEDRONS:         693
 dostot in rxesw  0.0000000E+00
#  BAND   13        
#EF=   0.52446     NDOS= 6     NENRG=  611    Gaussian bradening: 0.00300
 NUMBER OF ELECTRONS UP TO EF         :    5.3136

 the case.outputtdn file

 EMIN=  -0.19702 EMAX=   1.02446 EFACTR=    499.99996948 ESTEP =   0.00200

 ENERGY BAND    1 THROUGH   13 ENERGY CHANNEL:    1   TO  611
 NUMBER OF K-POINTS:         165
 NUMBER OF TETRAHEDRONS:         693
 dostot in rxesw  0.0000000E+00
#  BAND   13        
#EF=   0.52446     NDOS= 6     NENRG=  611    Gaussian bradening: 0.00300
 NUMBER OF ELECTRONS UP TO EF         :    4.6985

M = 5.3136 - 4.6985 = 0.6151