Lesson about the spin¶

This lesson aims at showing how to get the following physical properties:

• the total magnetization of a ferromagnetic material
• the magnetization of an antiferromagnetic material
• analyse the total density of states per spin direction
• analyse the density of states per atom and per spin direction
• look at the effect of spin-orbit coupling for a non magnetic system
• non-collinear magnetism (not yet)
• spin-orbit coupling and magnetocristalline anisotropy (not yet)

You will learn to use of features of ABINIT which deal with spin.

This lesson should take about 1.5 hour.

1 A ferromagnetic material: bcc Fe¶

Before beginning, you might consider to work in a different subdirectory as for the other lessons. Why not “Work_spin” (so ~abinit/tests/tutorial/Input/Work_spin)?

The file ~abinit/tests/tutorial/Input/tspin_x.files lists the file names and root names. You can copy it in the Work_spin directory (and change it, when needed, as usual).
You can also copy the file ~abinit/tests/tutorial/Input/tspin_1.in in Work_spin.

This is your input file. You can run the calculation, then you should edit the input file, and read it carefully.
Because we are going to perform magnetic calculations, there a two new types of variables:

Please, read their description in the help file.
You will work at fixed ecut (=18Ha) and k-point grid, defined by ngkpt (the 4x4x4 Monkhorst-Pack grid). It is implicit that in real life, you should do a convergence test with respect to both convergence parameters.
This takes about 12 seconds on a modern PC (one needs a sufficiently large cut-off to exhibit magnetic effects).

We will compare the output with and without magnetization. (Hence, there are two datasets in the run)
We now look at the output file.
In the magnetic case, the electronic density is split in two parts, the “Spin- up” and the “Spin-down” parts to which correspond different Kohn-Sham potentials and different sets of eigenvalues whose occupation are given by the Fermi-Dirac function (without the ubiquitous factor 2)

For the first k-point, we get for instance:

(no magnetization)
occ   2.00000   1.99989   1.99989   1.22915   1.22915   0.28676   0.00000   0.00000
(magnetic case)
occ   1.00000   0.99999   0.99999   0.98396   0.98396   0.69467   0.00000   0.00000 (spin-up)
1.00000   0.99730   0.99730   0.00898   0.00898   0.00224   0.00000   0.00000 (spin-down)


We note that the occupation are very different for up and down spins, which means that the eigenvalues are shifted which is in turn due to a shift of the exchange-correlation potential, and therefore of the effective potential. You can indeed have a look at the output file to compare spin-up and down eigenvalues:

-0.48411  -0.38615  -0.38615  -0.30587  -0.30587  -0.27293   0.33747   0.33747 (up, kpt#1)
-0.46638  -0.32383  -0.32383  -0.21767  -0.21767  -0.20371   0.36261   0.36261 (dn, kpt#1)


The magnetization density (in unit of mu_B - Bohr’s magneton) is the difference between the up and down densities. The magnetization density, divided by the total density is noted “zeta”. This quantity “zeta” can vary between -1 and 1. It is everywhere zero in the non-magnetic case. In the magnetic case, we can read for instance its minimal and maximal values in the output file:

Min spin pol zeta= -4.8326E-02 at reduced coord.  0.7222  0.5000  0.2222
next min= -4.8326E-02 at reduced coord.  0.5000  0.7222  0.2222
Max spin pol zeta=  5.7306E-01 at reduced coord.  0.0000  0.8889  0.8889
next max=  5.7306E-01 at reduced coord.  0.8889  0.0000  0.8889


The total magnetization, its integral on the unit cell, is now in the magnetic case:

Magnetization (Bohr magneton)=  1.96743463E+00
Total spin up =  4.98371731E+00   Total spin down =  3.01628269E+00


We observe that the total density (up+down) yields 8.000 as expected.

The magnetization density is not the only changed quantity. The energy is changed too, and we get:

etotal1  -2.4661707268E+01  (no magnetization)
etotal2  -2.4670792868E+01  (with magnetization)


The energy of the magnetized system is the lowest and therefore energetically favoured, as expected since bcc iron is a ferromagnet.
Finally, one notes also that the stress tensor is affected by the magnetization. This would also be true for the forces for a less symmetric material.

It is interesting to consider in more details the distribution of eigenvalues for each direction of magnetization, which is best done by looking at the respective densities of state.
To this end we have set prtdos= 1 in the input file, in order to obtain the density of states corresponding to spin-up and spin-down electrons (as soon as nsppol=2).
The values of the DOS are in the files tspin_1o_DS1_DOS and tspin_1o_DS2_DOS for the magnetic and non magnetic case respectively. We can extract the values for use in a plotting software.
Traditionally, in order to enhance visibility, one affects with a negative sign the DOS of minority electrons. If we compare the DOS of the magnetized] system

and the non magnetized system

we observe that the up and down DOS have been “shifted” with respect each other.
The integrated density of states yields the number of electrons for each spin direction and we see the magnetization which arises from the fact that there are more up than down electrons at the Fermi level.

That the magnetization points upwards is fortuitous, and we can get it pointing downwards by changing the sign of the initial spinat.
Indeed, in the absence of spin-orbit coupling, there is no relation between the direction of magnetization and the cristal axes.
If we start with a spinat of 0, the magnetization remains 0. spinat gives indeed a way to initially break the spin symmetry.

2 An antiferromagnetic example: fcc Fe¶

Well sort of....

Actually, fcc Fe, displays many complicated structures, in particular spin spirals. A spiral is characterized by a direction along an axis, an angle of the magnetization with respect to this axis and a step after which a rotation is complete.
A very simple particular case is when the angle is 90°, the axis is <100> and the step is the unit cell side, spin directions are alternating in every two planes perpendicular to the <100> axis (“spiral stairway”)
For instance, if the atom at [x,y,0] possesses an “up” magnetization, the atom at [x,y,½] would possess a down magnetization etc…
To describe such a structure, we merely need two atoms, [0,0,0] and [½,0,½] and a unit cell comprising those two atoms.
Also, each atom will be given opposite magnetization with the help of the variable spinat.

You can copy the file ~abinit/tests/tutorial/Input/tspin_2.in in Work_spin. This is your input file. Modify accordingly the tspin_x.files file.

You can run the calculation, then you should edit the tspin_2.in file, and briefly look at the two changes with respect to the file tspin_1.in: the change of unit cell basis vectors rprim, the new spinat.

Note also we use now nsppol=1 and nspden=2: this combination of values is only valid when performing an antiferromagnetic calculation (nspden=2 means that we have 2 independent components for the charge density while nsppol=1 means that we have 1 independent component for the wave-functions).
In that case, ABINIT uses the so-called Shubnikov symmetries, to perform calculations twice faster than with nsppol=2 and nspden=2. The symmetry of the crystal is not the full fcc symmetry anymore since the symmetry must now preserve the magnetization of each atom. ABINIT is nevertheless able to detect such symmetry belonging to the Shubnikov groups and correctly detects that the cell is primitive, which would not be the case if we had the same value of spinat on each atom.

If we now run again the calculation, this total computation time is approximately 30 seconds on a recent CPU.
If we look at the eigenvalues and occupations, they are again affected by a factor 2, which results from the symmetry considerations alluded to above, and not from the “usual” spin degeneracy: the potential for spin-up is equal to the potential for spin-down, shifted by the antiferromagnetic translation vector. Eigenenergies are identical for spin-up and spin-down, but wavefunctions are shifted one with respect to the other.

kpt#   1, nband= 16, wtk=  0.05556, kpt=  0.0833  0.0833  0.1250 (reduced coord)
-0.60539  -0.47491  -0.42613  -0.39022  -0.35974  -0.34377  -0.28895  -0.28828
-0.25314  -0.24042  -0.22943  -0.14218   0.20264   0.26203   0.26641   0.62158
occupation numbers for kpt#   1
2.00000   2.00000   2.00000   1.99997   1.99945   1.99728   1.50632 1.48106
0.15660   0.04652   0.01574   0.00000   0.00000   0.00000   0.00000 0.00000


How do we know we have a magnetic order?
The density of state used for bcc Fe will not be useful since the net magnetization is zero and we have as many up and down electrons.
Though, the magnetization is reflected in the existence of an up and down electronic density whose sum is the total density and whose difference yields the net magnetization density at each point in real space.

In particular, the integral of the magnetization around each atom will give an indication of the magnetic moment carried by this particular atom. A first estimation is printed out by ABINIT. You can read:

Integrated total density in atomic spheres:
-------------------------------------------
Atom  Sphere radius  Integrated_up_density  Integrated_dn_density
Total(up+dn)   Diff(up-dn)
1        2.00000             3.32789225             2.99093607    6.31882832    0.33695617
2        2.00000             2.98670724             3.32364305    6.31035029   -0.33693581
Note: Diff(up-dn) can be considered as a rough approximation of a local magnetic moment.


and obtain a rough estimation of the magnetic moment of each atom (strongly dependent on the radius used to project the charge density):

magnetization of atom 1= 0.33696
magnetization of atom 2=-0.33693


But here we want more precise results…
To perform the integration, we will use the utility cut3d which yields an interpolation of the magnetization at any point in space. cut3d is one of the executables of the ABINIT package and should be installed together with abinit.
As of now cut3d is interactive, we will use it through a very primitive script (written in Python) to perform a rough estimate of the magnetization on each atom.
You can have a look at the ~abinit/doc/tutorial/documents/lesson_spin/magnetization.py program, and note (or believe) that it does perform an integration of magnetization on a cube of side acell/2 around each atom; if applicable, you might consider to adjust the value of “CUT3D” string in the Python script.

Copy it in your Work_spin directory. If you run the program, by typing “python magnetization.py” , you will see the result:

magnetization of atom 1= 0.38212
magnetization of atom 2=-0.38203


which shows that the magnetization of each atom are actually opposite.
With the next input file tspin_3.in, we will consider this same problem, but in a different way, and note, for future record that the total energy is: Etotal=-4.92489592898935E+01

3 Another look at fcc Fe¶

Instead of treating fcc Fe directly as an antiferromagnetic material, we will not make any hypotheses on its magnetic structure, and run the calculation like the one for fcc Fe, anticipating only that the two spin directions are going to be different.
We will not even assume that the initial spins are of the same magnitude.

You can copy the file ~abinit/tests/tutorial/Input/tspin_3.in in Work_spin. This is your input file. You can modify the file tspin_x.files and immediately start running the calculation. Then, you should edit it.

Note the values of spinat. In this job, we wish again to characterize the magnetic structure.
We are not going to use zeta as in the preceding calculation, but we will here use another feature of abinit: atom and angular momentum projected densities of state.
These are densities of state weighted by the projection of the wave functions on angular momentum channels (that is spherical harmonics) centered on each atom of the system.
Note that theses DOS are computed with the tetrahedron method, which is rather time consuming and produces less smooth DOS than the smearing method. The time is strongly dependent on the number of kpoints and we use here only a reduced set.
(This will take about 1.5 minutes on a modern computer)

To specify this calculation we need new variables, in addition to prtdos set now to 3:

This will specify the atoms around which the calculation will be performed, and the radius of the sphere.
We specifically select a new dataset for each atom, a non self-consistent calculation being run to generate the projected density of states.
First, we note that the value of the energy is: Etotal=-4.92489557316370E+01, which shows that we have attained presumably the same state as above.

The density of states will be in the files tspin_3o_DS2_DOS_AT0001 for the first atom, and tspin_3o_DS3_DOS_AT0002 for the second atom.
We can extract the density of d states, which carries most of the magnetic moment and whose integral up to the Fermi level will yield an estimate of the magnetization on each atom. We note the Fermi level (recalled in the file tspin_3o_DS1_DOS):

Fermi energy :      -0.28270392


If we have a look at the integrated site projected density of states, we can compute the total moment on each atom. To this end, one can open the file tspin_3o_DS3_DOS_AT0002, which contains information pertaining to atom 2. This file is self-documented, and describes on each line, for spin up and spin down:

# energy(Ha)  l=0   l=1   l=2   l=3   l=4    (integral=>)  l=0   l=1   l=2 l=3   l=4


If we look for the lines containing an energy of “-0.28250”, we find

up -0.28250 0.8026 2.6082 23.3966 0.7727 0.1687 0.30 0.34 3.42 0.04 0.01
dn -0.28250 0.3381 1.8716 24.0456 0.3104 0.1116 0.30 0.33 2.74 0.04 0.01

There are apparently changes in the densities of states for all the channels, but besides the d-channels, these are indeed fluctuations. This is confirmed by looking at the integrated density of states which is different only for the d-channel. The difference between up and down is 0.68, in rough agreement (regarding our very rude methods of integration) with the previous calculation. Using a calculation with the same number of k points for the projected DOS, we can plot the up-down integrated dos difference for the d-channel.

Note that there is some scatter in this graph, due to the finite number of digits (2 decimal places) of the integrated dos given in the file tspin_3o_DS3_DOS_AT0002.

If we now look at the up and down DOS for each atom, we can see that the corner atom and the face atom possess opposite magnetizations, which roughly cancel each other. The density of states computed with the tetrahedron method is not as smooth as by the smearing method, and a running average allows for a better view.

4 Ferrimagnetic (not yet)¶

Some materials can display a particular form of ferromagnetism, which also can be viewed as non compensated antiferromagnetism, called ferrimagnetism.
Some atoms possess up spin and other possess down spin, but the total spin magnetization is non zero.
This happens generally for system with different type of atoms, and sometimes in rather complicated structures such as the magnetite.

5 The spin-orbit coupling¶

For heavy atoms a relativistic description of the electronic structure becomes necessary, and this can be accomplished through the relativistic LDA approximation.

5.1 Norm-conserving pseudo-potentials¶

For atoms, the Dirac equation is solved and the 2(2l+1) l-channel degeneracy is lifted according to the eigenvalues of the L+S operator (l+½ and l-½ of degeneracy 2l+2 and 2l). After pseudization, the associated wave functions can be recovered by adding to usual pseudo-potential projectors a spin-orbit term of the generic form $v(r).|l,s\rangle L.S \langle l,s|$. Not all potentials include this additional term, but it turns out that it is the case for all the HGH type pseudopotentials.

In a plane wave calculation, the wavefunctions will be two components spinors, that is will have a spin-up and a spin-down component but these components will be coupled. This mean the size of the Hamiltonian matrix is quadrupled.

We will consider here a heavier atom than Iron: Tantalum.
You will have to change the “files” file accordingly, as we here have used the potential: 73ta.hghsc. It is a HGH pseudopotential, with semicore states. Replace the last line of the tspin_x.files by

../../../Psps_for_tests/73ta.hghsc


You can copy the file ~abinit/tests/tutorial/Input/tspin_5.in in Work_spin. Change accordingly the file names in tspin_x.files, then run the calculation. It takes about 20 secs on a recent computer.

The input file contains one new variable:

Have a look at it. You should also look at so_psp; it is not used here, meaning that the SO information is directly read from the pseudopotential file.

In this run, we check that we recover the splitting of the atomic level by performing a calculation in a big box. Two calculations are launched with and without spin-orbit.

We can easily follow the symmetry of the different levels of the non spin orbit calculation:

  kpt#   1, nband= 26, wtk=  1.00000, kpt=  0.0000  0.0000  0.0000 (reduced coord)
-2.44760
-1.46437  -1.46437  -1.46437
-0.17045
-0.10852  -0.10852  -0.10852  -0.10740  -0.10740


That is, the symmetry: s, p, s, d
After application of the spin-orbit coupling, we now have to consider twice as many levels:

 kpt#   1, nband= 26, wtk=  1.00000, kpt=  0.0000  0.0000  0.0000 (reduced coord)
-2.43258  -2.43258
-1.67294  -1.67294  -1.35468  -1.35468  -1.35468  -1.35468
-0.16788  -0.16788
-0.11629  -0.11629  -0.11629  -0.11629  -0.09221  -0.09221 -0.09120  -0.09120  -0.09120  -0.09120


The levels are not perfectly degenerate, due to the finite size of the simulation box. We can nevetheless compute the splitting of the levels, and we obtain, for e.g. the p-channel: 1.67294-1.35468=0.31826 Ha

If we now consider the NIST table of atomic data, we obtain:

5p splitting, table: 1.681344-1.359740=0.321604 Ha
5d splitting, table:   .153395-.131684=0.021711 Ha


We obtain a reasonable agreement.
A more converged (and more expensive calculation) would yield:

5p splitting, abinit: 1.64582-1.32141=0.32441 Ha
5d splitting, abinit:   .09084-.11180=0.02096 Ha


5.2 Projector Augmented-Wave¶

Within the Projector Augmented-Wave method, the usual (pseudo-)Hamiltonian can be expressed as:

H = Kin + V_eff + Σ D_ij |~pi > <~pj |

If the two following conditions are satisfied:

(1) the local PAW basis is complete enough;
(2) the electronic density is mainly contained in the PAW augmentation regions,

it can be showed that a very good approximation of the PAW Hamiltonian – including spin-orbit coupling – is:

H ~ Kin + V_eff + Σ (D_ij+D^SO_ij) |~pi > <~pj |

where D^SO_ij is the projection of the (L.S) operator into the PAW augmentation regions.
As an immediate consequence , we thus have the possibility to use the standard ~pi PAW projectors; in other words, it is possible to use the standard PAW datasets (pseudopotentials) to perform calculations including spin-orbit coupling.
But, of course, it is still necessary to express the wave-functions as two components spinors (spin-up and a spin-down components).
Let’s have a look at the following keyword:

It automatically activates the spin-orbit coupling within PAW (forcing nspinor=2).

Now the practice:
We consider here Bismuth.
You will have to change the “files” file accordingly, as we here have used the potential: 83bi.paw. It is a PAW dataset with 5d, 6s and 6p electrons in valence.
Replace the last line of the tspin_x.files by

../../../Psps_for_tests/83bi.paw


You can copy the file ~abinit/tests/tutorial/Input/tspin_6.in in Work_spin (one Bismuth atom in a large cell). Change accordingly the file names in tspin_x.files, then run the calculation. It takes about 10 secs on a recent computer.

Two datasets are executed: the first one without spin-orbit coupling, the second one using pawspnorb=1.

The results (eigenvalues) are:

 Eigenvalues (hartree) for nkpt=   1  k points:
kpt#   1, nband= 24, wtk=  1.00000, kpt=  0.0000  0.0000  0.0000 (reduced
coord)
5d   -0.93353  -0.93353  -0.93353  -0.93353  -0.82304  -0.82304  -0.82304
-0.82304 -0.82291  -0.82291
6s   -0.42972  -0.42972
6p   -0.11089  -0.11089  -0.03810  -0.03810  -0.03810  -0.03810


Again, the levels are not perfectly degenerate, due to the finite size of the simulation box.
We can compute the splitting of the levels, and we obtain:

5d-channel: 0.93353-0.82304=0.11048 Ha
6p-channel: 0.11089-0.03810=0.07289 Ha


If we now consider the NIST table of atomic data, we obtain:

5d-channel: 1.063136-0.952668=0.11047 Ha
6p-channel: 0.228107-0.156444=0.07166 Ha


A perfect agreement even with a small simulation cell and very small values of plane-wave cut-offs…

6 Rotation of the magnetization and spin-orbit¶

The most spectacular manifestation of the spin-orbit coupling is the energy associated with a rotation of the magnetisation with respect with the crystal axis. It is at the origin of the magneto crystalline anisotropy of paramount technological importance.

GZ would like to thank B. Siberchicot for useful comments.