Magnetism in lmf

LDA description of ErAs

Erbium arsenide has the rock salt structure; the 11 Er $f$ electrons take up localised, atomic-like states and form a moment of $\sim3\mu_B$ (7 “up”, 4 “down”). The LDA description, failing to distinguish between the occupied and unoccupied minority f-states, places these minority states at $E_F$, 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

Edit the ctrl file to prepare for a magnetic calculation:

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

The calculation converges to a total moment $2.72\mu_B$ somewhat less than 3. Notice that the total moment is also echoed in the save.eras file. $n(E_F)~1000$ states/Ryd.

To plot the density of states:

The arguments to pldos limit the $n(E)$ 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:

Notes:

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

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

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

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.

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:

and,

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 $n(E_F)$ is sensible.

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 $L.S$ to the Hamiltonian. SO=2,3 are methods, based on $L_zS_z$, for keeping the Hamiltonian spin-diagonal as is necessary for the GW codes.

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 $k$-path might be (copy into qp.eras):

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