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Magnetism in lmf

LDA description of ErAs

Erbium arsenide has the rock salt structure; the 11 Er electrons take up localised, atomic-like states and form a moment of (7 “up”, 4 “down”). The LDA description, failing to distinguish between the occupied and unoccupied minority f-states, places these minority states at , giving a metallic description. LDA+U correctly splits the occupied and unoccupied states, moving the f-states away from the Fermi-level and restoring the observed semi-metallic behaviour.

1. Converged LDA calculation and density of states

cp /home/vol05/tmp15/eras_icsd_43644.cif .
cif2cell eras_icsd_43644.cif >out
cif2init out
mv init init.eras
blm eras
cp actrl.eras ctrl.eras
vi ctrl.eras

Edit the ctrl file to prepare for a magnetic calculation:

% const nit=50
% const so=0 nsp=2                 # spin polarisation
% const lxcf=1 lxcf1=0 lxcf2=0     # use CA not BH
% const nkabc=0 gmax=0 beta=.1     # small mixing for tricky system


ATOM=Er       Z= 68  R= 2.846545  LMX=3 LMXA=6 MMOM=0 0 0 3

Run the atom solver lmfa, and update the gmax token with the recommended value (9.4 a.u.). Set nkabc=8, a reasonable mesh here.

lmfa --usebasp eras | tee log_lmfa
grep GMAX log_lmfa
vi ctrl.eras
mpirun -np 24 lmf eras | tee log_lmf-lda

The calculation converges to a total moment somewhat less than 3. Notice that the total moment is also echoed in the save.eras file. states/Ryd.

 BZWTS : --- Tetrahedron Integration ---
         Est E_f           Window        Tolerance  DOS(E_f)
        -0.041188  -0.041424  -0.040460   0.000964 752.425776
        -0.041241  -0.041251  -0.041241   0.000010 **********
        -0.041241  -0.041241  -0.041241   0.000000 **********
 BZINTS: Fermi energy:     -0.041241;  25.000000 electrons;  D(Ef): 1025.571
         Sum occ. bands:  -14.1291269  incl. Bloechl correction:   -0.000399
         Mag. moment:       2.721098

 Saved qp weights ...

 mkrout:  Qtrue      sm,loc       local        true mm   smooth mm    local mm
   1   18.434947    4.536714   13.898233      2.623196    0.114373    2.508824
       contr. to mm extrapolated for r>rmt:   0.050070 est. true mm = 2.673266
   2    4.113469    4.620245   -0.506777      0.059428    0.057311    0.002117
       contr. to mm extrapolated for r>rmt:   0.082335 est. true mm = 0.141763

To plot the density of states:

mpirun -np 24 lmf --quit=dos --dos~npts=2001~window=-1,1 eras |tee log_dos
echo 25 20 -12 12 | pldos -esclxy=13.6057 -ef=0 -fplot -lst=1 -lst2 dos.eras
fplot -f plot.dos
ps2pdf fplot.ps eras_dos_lda.pdf

The arguments to pldos limit the scale to 25 states/eV , sets the figure size to 20cm and plots between -12 and 12 eV. Note “-lst2” which includes the second spin channel with opposite sign.

Of course the system is ferromagnetic by construction because there is only one Er site.


2. LDA+U calculation and density of states

Update the ctrl file by adding these tokens for the LDA+U method:

SYMGRP R4Z MZ

ITER
 UMIX=0.5

SPEC
  ATOM=Er       Z= 68  R= 2.846545  LMX=3 LMXA=6 MMOM=0 0 0 3
  IDU= 0 0 0 12
  UH=  0 0 0 {8.0/12.6057}
  JH=  0 0 0 {0.5/12.6057}

Notes:

  • IDU=0,1,2 for no +U, AMF or FLL double counting corrections – a list over
  • IDU+=10 flags the inclusion of as well as
  • UH,JH values of U,J in Ryd
  • UMIX stabilises the calculation (mixing beta for the density matrix)
  • remove symmetry operations which rotate
SYMGRP find SOC=1

(find symmetry operations for LDA+U or SO.)

Initial occupations can be specified in a simple way in the occnum.eras file:

% real
1 1 1 1 1 1 1
0 0 0 1 1 1 1

Notes:

  • without %real, complex spherical harmonics are understood
  • ordered -l:l for majority, minority spins
  • add blocks in sequence for more than one +U set

Run lmf as usual, repeat as required.

 sudmtu:  no file dmats ... read (diagonal) density-matrix from occnum file
 occ numbers, site 1 l=3:   1 1 1 1 1 1 1 (spin 1)   0 0 0 1 1 1 1 (spin 2)
 sudmtu:  RMS change in dmats from symmetrization = 0.071429
          (warning) RMS change unexpectely large

lmf prints the new density matrices at every iteration and updates the dmats.eras file (which is human-readable). Ultimately, lmf reports the converged results:

         LDA+U total energy ...
 ldau:   version = 2  iblu = 1
 vldau:  Eldau = 20.860170  Edc = 20.330652  Eorb = 0.529517
         eks = -30673.344939  e[U] = 0.529517  Etot(LDA+U) = -30672.815422
         LDA+U update density matrix ...
         RMS diff in dens mat(1.22e-4) > tolu (0) Linear mix with beta=0.5
         RMS change in vorb from symmetrization = 0.000003
 Mixed dmats l=3  site 1  spin 1, spherical harmonics
  0.78743    0.00000    0.00016    0.00000    0.00000    0.00000    0.00000
  0.00000    0.78619    0.00000    0.00000    0.00000   -0.00008    0.00000
  0.00016    0.00000    0.79146    0.00000    0.00000    0.00000    0.00000
  0.00000    0.00000    0.00000    0.78855    0.00000    0.00000    0.00000
  0.00000    0.00000    0.00000    0.00000    0.79186    0.00000   -0.00016
  0.00000   -0.00008    0.00000    0.00000    0.00000    0.78917    0.00000
  0.00000    0.00000    0.00000    0.00000   -0.00016    0.00000    0.78971

  0.00000    0.00000    0.00000    0.00000   -0.00001    0.00000    0.00000
  0.00000    0.00000    0.00000    0.00000    0.00000   -0.00002    0.00000
  0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00001
  0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000
  0.00001    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000
  0.00000    0.00002    0.00000    0.00000    0.00000    0.00000    0.00000
  0.00000    0.00000   -0.00001    0.00000    0.00000    0.00000    0.00000
 Mixed dmats l=3  site 1  spin 2, spherical harmonics
  0.01420    0.00000    0.00094    0.00000    0.00367    0.00000    0.00000
  0.00000    0.00464    0.00000    0.00000    0.00000   -0.00439    0.00000
  0.00094    0.00000    0.01646    0.00000    0.00000    0.00000    0.00367
  0.00000    0.00000    0.00000    0.76925    0.00000    0.00000    0.00000
  0.00367    0.00000    0.00000    0.00000    0.77357    0.00000   -0.00094
  0.00000   -0.00439    0.00000    0.00000    0.00000    0.76978    0.00000
  0.00000    0.00000    0.00367    0.00000   -0.00094    0.00000    0.76992

  0.00000    0.00000   -0.00022    0.00000   -0.00002    0.00000    0.00000
  0.00000    0.00000    0.00000    0.00000    0.00000    0.00002    0.00000
  0.00022    0.00000    0.00000    0.00000    0.00000    0.00000    0.00002
  0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000
  0.00002    0.00000    0.00000    0.00000    0.00000    0.00000   -0.00022
  0.00000   -0.00002    0.00000    0.00000    0.00000    0.00000    0.00000
  0.00000    0.00000   -0.00002    0.00000    0.00022    0.00000    0.00000

and,

IORBTM:  orbital moments :
 site  spec        spin   Moment decomposed by l ...
    1    Er           1    0.000000   -0.012462   -0.031719    0.008341    0.000126    0.000034    0.000013
    1    Er           2    0.000000    0.012755    0.028980    5.917473   -0.000126   -0.000039   -0.000011
    1  L+ - L-   5.923366
    2    As           1    0.000000   -0.020485    0.001009   -0.000002    0.000000    0.000000    0.000000
    2    As           2    0.000000    0.018723   -0.001704   -0.000234    0.000000    0.000000    0.000000
    2  L+ - L-  -0.002693

 BZWTS : --- Tetrahedron Integration ---
         Est E_f           Window        Tolerance  DOS(E_f)
        -0.003115  -0.003141  -0.002077   0.001064   2.351381
        -0.003115  -0.003120  -0.003109   0.000011   2.347617
        -0.003115  -0.003115  -0.003115   0.000000   2.347612
 BZINTS: Fermi energy:     -0.003115;  25.000000 electrons;  D(Ef):    2.348
         Sum occ. bands:  -19.5837013  incl. Bloechl correction:   -0.000834
         Mag. moment:       2.995678

 Saved qp weights ...

 mkrout:  Qtrue      sm,loc       local        true mm   smooth mm    local mm
   1   18.322237    5.281827   13.040410      3.005389    0.176453    2.828936
 mag true  -0.000054  0.000037  3.005389  smooth -0.000143 -0.000268  0.176452
       contr. to mm extrapolated for r>rmt:   0.007134 est. true mm = 3.012523
   2    4.148275    4.658113   -0.509838      0.017586    0.021715   -0.004129
 mag true  -0.000104 -0.000233 -0.017584  smooth -0.000092 -0.000155 -0.021714
       contr. to mm extrapolated for r>rmt:  -0.001736 est. true mm = 0.015850

Corresponding to the occnum.eras, we arrived at the configuration with maximum orbital moment. Other configurations, with different energies, can be reached from different starting configurations (try!).

Both the total and Er local moment are close to 3, and is sensible.

mpirun -np 24 lmf --quit=dos --dos~npts=2001~window=-1,1 eras |tee log_dos
echo 25 20 -12 12 | pldos -esclxy=13.6057 -ef=0 -fplot -lst=1 -lst2 dos.eras
fplot -f plot.dos
ps2pdf fplot.ps eras_dos_ldau.pdf

LDA+U is quite effective here.


3. Adding spin-orbit and plotting the band structure

Including spin-orbit is achieved by setting HAM_SO greater than zero. SO=1 adds the full to the Hamiltonian. SO=2,3 are methods, based on , for keeping the Hamiltonian spin-diagonal as is necessary for the GW codes.

% const so=1 nsp=2

Because the spin channels are now mixed, the plotting spin-up and -down bands doesn’t make sense. Instead it is effective to colour the weights by the spin: here we highlight the Er majority f-states in red, minority in green. A suitable -path might be (copy into qp.eras):

41  .5 .5 .5     0  0 0                L to Gamma   (Lambda)
41   0  0  0     1  0 0                Gamma to X   (Delta)
21   1  0  0     1 .5 0                X to W       (Z)
41   1 .5  0     0  0 0                W to Gamma   (no name?)
0    0 0 0  0 0 0

then we proceed as usual, noting that the spin-down basis functions are offset by 48 from the spin-up.

lmf --pr61 --quit=ham eras
lmf --band~col=10:16,26:32~col2=48+10:48+16,48+26:48+32 eras
echo -10 10 10 10 | plbnds -fplot -ef0 -scl=12.6057 -lbl=L,G,X,W,G -lt=1,col=0,0,0,colw=1,0,0,colw2=0,1,0 bnds.eras
fplot -f plot.plbnds
ps2pdf fplot.ps eras_bands_ldaso.pdf

ErAs bands LDA+U