Calculating the hyperfine coupling constant
The hyperfine coupling constant (cf. hyperfine splitting) describes the interaction between the nuclear magnetic dipole moment and the magnetic field generated by the electrons (i.e. the nuclear spin-electron spin coupling. The hyperfine coupling constant is calculated using LHYPERFINE [1]. The hyperfine splitting often includes the interaction between the nuclear quadrupole moment and the electric field gradient (EFG), which is calculated separately using LEFG and the description in performing an EFG calculation. The hyperfine splitting can be measured using electron paramagnetic resonance (EPR), also called electron-spin resonance (ESR), and in atomic spectroscopy. The theory is covered in the NMR category page and LHYPERFINE page.
Step-by-step instructions
The hyperfine constant is calculated post-self-consistent field (SCF) using LHYPERFINE. A well-converged SCF calculation is therefore crucial. The hyperfine coupling constant is sensitive to several input parameters that must all be tested.
Step 1 (optional): Calculate the hyperfine constant using a previously converged calculation. Since the hyperfine constant is calculated post-SCF, you can use a previously converged CHGCAR to calculate it alongside LCHARG.
Step 2: Define the nuclear gyromagnetic rations The hyperfine coupling constant is dependent on the nuclear gyromagnetic ratios defined in NGYROMAG. Since the defaults are set to 1, the gyromagnetic rations must be defined for each element in your POSCAR file to obtain meaningful coupling constants.
Step 3: Converge with respect to the plane-wave energy cutoff The plane-wave basis can strongly influence the coupling constant. Unconverged values are not meaningful and should not be compared to experiment. Multiple calculations should be performed while increasing the basis set size, as defined in ENCUT, incrementally (e.g., by 100 eV intervals). Once the value of the coupling constant no longer changes to within 0.1 MHz, it is converged.
'Step 4: Converge with respect to the k point mesh Similar to the basis, the k' point mesh can strongly influence the coupling constant. The k point mesh should be increased incrementally (EVEN NUMBERED MESHES???) until convergence to within 0.1 MHz is achieved.
Step 5: Compare to experiment The Fermi contact term can be found and compared to values determined in experiment CITE_ADAM_GALI_SZASZ.
Input
The hyperfine coupling constants are calculated using LHYPERFINE
There is one additional keyword that must be defined:
- NGYROMAG defines the nuclear gyromagnetic ratios for each element in your POSCAR file. The defaults are set to 1, which will return meaningless results. Reasonable values may be found here.
An example INCAR file is given here:
ENCUT = 500 # Plane-wave energy cutoff in eV ISMEAR = 0; SIGMA = 0.01 # Defines the type of smearing; smearing width in eV EDIFF = 1E-6 # Energy cutoff criterion for the SCF loop, in eV PREC = Accurate # Sets the "precision" mode LHYPERFINE = .TRUE. # Turns on calculating the hyperfine coupling tensor NGYROMAG = 10.7084 42.577478461 # Specifies the nuclear gyromagnetic ratios for the ions - C and H in this case ISPIN = 2 # Turns on spin-polarization - noncollinear can also be used
| Important: Make sure to replace the NGYROMAG in the INCAR with the values for the isotopes in your system. |
Output
You can find the output for the hyperfine calculation in the OUTCAR file after the SCF cycle finishes. The total magnetic moment is listed, then the Fermi contact term:
Total magnetic moment S= 2.0000000 Fermi contact (isotropic) hyperfine coupling parameter (MHz) ------------------------------------------------------------- ion A_pw A_1PS A_1AE A_1c A_tot ------------------------------------------------------------- 1 - - - - - 2 - - - - - -------------------------------------------------------------
Note the A_tot does not include the core contribution term A_1c [2]. The dipolar hyperfine coupling parameter comes next and finally the total hyperfine coupling parameter
Dipolar hyperfine coupling parameters (MHz) --------------------------------------------------------------------- ion A_xx A_yy A_zz A_xy A_xz A_yz --------------------------------------------------------------------- 1 - - - - - - 2 - - - - - - --------------------------------------------------------------------- Total hyperfine coupling parameters after diagonalization (MHz) (convention: |A_zz| > |A_xx| > |A_yy|) ---------------------------------------------------------------------- ion A_xx A_yy A_zz asymmetry (A_yy - A_xx)/ A_zz ---------------------------------------------------------------------- 1 - - - - 2 - - - - ---------------------------------------------------------------------
Recommendations and advice
The hyperfine constant is less dependent on EDIFF and ENCUT, generally converging relatively quickly with respect to both. However, it is extremely strongly influenced by the method used. HSE06 was found to give values close to experimental values [1]. Additionally, the choice of k-point mesh KPOINTS used can be very important.
- The PAW pseudopotential that you use should be considered.
- Including additional electrons in the valence (i.e. _sv, _pv) can be important.
- GW pseudopotentials (i.e. _GW) can have a big effect on the hyperfine coupling parameter.
- Hybrid functionals give a better description than GGA. HSE06 can localize defect states, improving their description relative to PBE [1].
- We recommend using tightly converged settings:
PREC = Accurate EDIFF = 1E-6 # Note that some systems might require tighter settings, e.g. 1E-8
- Increase the plane-wave energy cutoff ENCUT and k-point mesh KPOINTS until convergence has been achieved for the hyperfine coupling parameter.
- If increasing the k-point mesh causes the coupling to disappear (all zeros), this is an indicator that you have a non-magnetic solution.
- If your system relaxes to a non-magnetic solution and you think that it should be magnetic, you can enforce it using NUPDOWN to specify the number of unpaired electrons. Carefully check that the energies are lower for the magnetic solution.
- Test your system with LASPH = .TRUE. and .FALSE. In some cases, non-spherical contributions may be important.
References
- ↑ a b c K. Szasz, T. Hornos, M. Marsman, and A. Gali, Hyperfine coupling of point defects in semiconductors by hybrid density functional calculations: The role of core spin polarization, Phys. Rev. B, 88, 075202 (2013).
- ↑ O. V. Yazyev, I. Tavernelli, L. Helm, and U. R. Roethlisberger, Core spin-polarization correction in pseudopotential-based electronic structure calculations, Phys. Rev. B 71, 115110 (2006).