# Zeeman Field Tutorial

Note: this page has not been carefully worked out, especially the description of the Stoner parameter may not be accurate.

You can add to the crystal potential an external magnetic field inside each augemetation sphere, in the ASA program **lm** and full-potential program **lmf**. Use token BFIELD to tell **lm** or **lmf** that a field is to be read. The calculation must be spin polarized, e.g.

```
HAM NSPIN=2 BFIELD=2
...
```

*BFIELD=1* allows the field to point in any direction; *BFIELD=2* restricts the field to the *z* axis. The former case generates a noncollinear hamiltonian; the latter is spin diagonal. (The hamiltonian may also be noncollinear for another reason, e.g. spin-orbit coupling is turned on.) Caution: **lmf** does not as yet generate noncollinear output densities. Self-consistent calculations with noncollinear **B** fields are not possible; also the total energy is not trustworthy if **B** is not along the *z* axis.

By selecting BFIELD nonzero, you are telling **lm** or **lmf** that a (site-dependent) Zeeman field is to be added. In this case you must also supply a site-dependent external field, which you do through the additional file bfield.ext. If this file is omitted, the program will not execute. bfield.ext contains one row for each site; each row contains three numbers corresponding to *x*,*y*,*z* components of **B**. (The file must always contain three components *B _{x}.*

*B*

_{y}*B*even if

_{z}*BFIELD=2*.) The GdN test case (part of the test suite) is an instructive example. In the top-level directory, run this test:

```
fp/test/test.fp gdn
```

This test combines the LDA+*U* hamiltonian with spin-orbit coupling. Bands are calculated for that case; then a magnetic field of 0.25 eV=0.037/2 Ry is added to the Gd site and the bands are recalculated. GdN has two atoms and file *bfield.gdn* reads:

```
0 0 .037/2
0 0 0
```

The test generates files *bnds.bfield.gdn* and *bnds.nobfield.gdn* To analyze the bands you need to use a graphics package. If you have the FPLOT packaged installed, you can draw the picture shown below comparing the two cases, by invoking the following commands after running the test:

```
echo -8,8,5,10 | plbnds -fplot -ef=0 -scl=13.6 -lbl=L,G,X,W,G -dat=blue bnds.nobfield.gdn
echo -8,8,5,10 | plbnds -fplot -ef=0 -scl=13.6 -lbl=L,G,X,W,G -dat=dat bnds.bfield.gdn
mv plot.plbnds plot.plbnds~
echo "% char0 colr=3,bold=4,clip=1,col=1,.2,.3" >>plot.plbnds
echo "% char0 colb=2,bold=2,clip=1,col=.2,.3,1" >>plot.plbnds
awk '{if ($1 == "-colsy") {sub("-qr","-lt {colr} -qr");print;sub("dat","blue");sub("colr","colb");print} else {print}}' plot.plbnds~ >>plot.plbnds
fplot -disp -f plot.plbnds
```

In the figure, blue bands are LDA+*U* results. Because spin-orbit coupling is included both minority and majority spin bands appear. You can see majority Gd *f* states near -6.5 eV, and minority *f* states near +5.5 eV. Gd *d* states are spin-split by the exchange-correlation field originating from spin splitting of the Gd *f* states; majority and minority *d* states are found near 2.5 eV and 3.5 eV, respectively, at the Γ point.

Red bands use the same potential as the LDA+*U* case, except that a *B _{z}* field of magnitude 0.5 eV was added to the Gd site.

*B*causes the the following shifts:

_{z}- The majority
*f*shifts*down*by ~0.22 eV, and the minority*f*states shift*up*by ~0.27 eV. States that are purely atomic like and are little affected by the kinetic energy will be shifted by the Zeeman field, ±*B*/2. This is the case for Gd*f*orbitals. - The majority
*d*states at Γ shift*down*by ~0.16 eV; the minority*d*states shift*up*by ~0.20 eV. - The As
*p*states, in the (-4,0) eV range, are only slightly shifted.

### Longitudinal susceptibility

The magnetic susceptibility is the system’s magnetic response to an external magnetic field:

$\delta M = \chi \delta B$or

$\delta B = \chi^{-1} \delta M$δM itself can be obtained from a derivative of the total energy:

$\delta M = {\delta E\over \delta B} \delta B$Thus

$\chi = {\delta^2 E \over \delta B^2}$and

$\chi^{-1} = {\delta^2 E\over \delta M^2}$In linear response theory, $\chi^{-1}$ can be written as a sum of a noninteracting part and an interacting part

$\chi^{-1} = \chi_{0}^{-1} - I$The kernel $I$ is often associated with the “Stoner parameter.” It is typically about 1 eV in 3*d* transition metals. The noninteracting part $\chi_0^{-1}$ is connected with the induced magnetization of the noninteracting system, i.e.

where $\delta M_0$ is the magnetization induced by $\delta B$ without taking into account the internal potential shifts that $\delta M$ cause. In the density-functional context, $\delta M_0$ can be obtained from an initially self-consistent potential $V_0$, which is generated in the absence of an external field. $\delta M_0$ is then the change in $M$ for the potential $V_0 + \delta B$. In practice this is obtained by evaluating the change in $M$ from the change in output density of a 1-shot calculation with potential $V_0 + \delta B$.

You can use **lmf** to apply a field and calculate *E_LDA and δ_M* as a function of B. By comparing self-consistent and 1-shot calculations you can in principle also obtain *I*. The example below does this in an approximate way for Fe, using input file fp/test/ctrl.felz. Note: a proper calculation of *I* would require a supercell large enough that the interactions between external *B*-fields would be sufficiently small. (*I* is a local function of **r** in density functional theory; thus *I* is a site-diagonal matrix.) Here we only consider a single atom cell. The interactions are not small, and *I* is underestimated. The results below were generated by this command:

```
lmf -vnk=16 --pr31 -vrel=1 -vnit=50 -vso=0 --rs=1 -vbeta=.3 -vbf=2 -vbz=$bz -vconv=1d-6 -vconvc=1d-6 felz
```

bz must be set as a shell variable.

Magnetization and total energy (relative to the B=0 case) were generated for several values of B, and ∂E/∂B was obtained from numerical differentiation of *E*.

As the top figure shows, the total energy *E* has a minimum near, but not exactly at B=0. This shows that there are some slight errors in the LDA calculation (in an exact self-consistent LDA calculation *E* should be minimum at exactly B=0). *E* is smooth in the interval (−0.01, 0.01) Ry, as the figure shows. The behavior is somewhat unsmooth near B=−0.01 Ry, and its discontinuity or near discontinuity is evident near B=0.01 Ry. That means numerical differentiation ∂E/∂B is a bit problematic, particularly outside the (−0.01,0.01) interval. Nevertheless we can perform it. In the bottom figure, *M*−*M*(B=0) and ∂E/∂B are plotted respectively as a function of B as dashed lines+circles, and a solid line. The two curves track fairly well, though ∂E/∂B is noisy; nor is *M*(B) particularly smooth.

We can obtain χ from the slope ∂*E*/∂B. There is some uncertainty in the calculation since *M* is not particularly smooth (note in particular that (∂E/∂B)_{+} and (∂E/∂B)_{−} are a little different).

We can also obtain both χ_{0} by obtaining δM_{0} from a 1-shot calculation. The table below shows M_{0} and *M* for two values of B:

B | M_{0} | M |
---|---|---|

-0.005 | -2.363587 | -2.402973 |

0.005 | -2.234213 | -2.219662 |

Thus we get:

$\chi_{0}^{-1} = {0.01\over -2.234213 - -2.363587} = 0.0773 Ry \Rightarrow \chi_{0} = 12.9 Ry^{-1} = 0.95 eV.$ $\chi^{-1} = {0.01 \over -2.219662 - -2.402973} = 0.0546 Ry \Rightarrow \chi = 18.3 Ry^{-1} = 1.35 eV.$ $I = \chi_{0}^{-1} - \chi^{-1} = 0.0227 Ry = 0.30 eV$As noted above, *I* calculated in this way only approximately corresponds to the Stoner *I*. A better calculation should produce *I* about 1 eV.

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