LHYPERFINE

LHYPERFINE = .TRUE. | .FALSE.
Default: LHYPERFINE = .FALSE.

Description: compute the hyperfine tensors at the atomic sites (available as of vasp.5.3.2).

To have VASP compute the hyperfine tensors at the atomic sites, set

LHYPERFINE = .TRUE.

Mind: Either spin-polarized calclulations ISPIN = 2 or noncollinear calculations LNONCOLLINEAR = .TRUE. must be used.

The hyperfine tensor A^I describes the interaction between a nuclear spin S^I (located at site R_I) and the electronic spin distribution S^e (in most cases associated with a paramagnetic defect state):

{\displaystyle E=\sum_{ij} S^e_i A^I_{ij} S^I_j }

In general it is written as the sum of an isotropic part, the so-called Fermi contact term, and an anisotropic (dipolar) part.

The Fermi contact term is given by

{\displaystyle (A^I_{\mathrm{iso}})_{ij}= \frac{2}{3}\frac{\mu_0\gamma_e\gamma_I}{\langle S_z\rangle}\delta_{ij}\int \delta_T(\mathbf{r})\rho_s(\mathbf{r}+\mathbf{R}_I)d\mathbf{r} }

where ρ_s is the spin density, μ₀ is the magnetic susceptibility of free space, γ_e the electron gyromagnetic ratio, γ_I the nuclear gyromagnetic ratio of the nucleus at R_I, and $\langle S_z \rangle$ the expectation value of the z-component of the total electronic spin.

δ_T(r) is a smeared out δ function, as described in the Appendix of Ref. ^[1].

The dipolar contributions to the hyperfine tensor are given by

{\displaystyle (A^I_{\mathrm{ani}})_{ij}=\frac{\mu_0}{4\pi}\frac{\gamma_e\gamma_I}{\langle S_z\rangle} \int \frac{\rho_s(\mathbf{r}+\mathbf{R}_I)}{r^3}\frac{3r_ir_j-\delta_{ij}r^2}{r^2} d\mathbf{r} }

In the equations above r=|r|, r_i the i-th component of r, and r is taken relative to the position of the nucleus R_I.

The nuclear gyromagnetic ratios should be specified by means of the NGYROMAG-tag:

NGYROMAG = gamma_1  gamma_2 ... gamma_N

where one should specify one number for each of the N species on the POSCAR file, i.e. if C, H, N, and O are listed as species in the POSCAR file, then there should be four numbers in NGYROMAG, regardless of how many total atoms there are.

Important: If one does not set NGYROMAG in the INCAR file, VASP assumes a factor of 1 for each species.

Input

An example INCAR is presented below for a Nitrogen-vacancy (NV) center in diamond:

 PREC = Accurate  
 ENCUT = 500         
 EDIFF = 1E-8  
 ISMEAR = 0; SIGMA = 0.01 

 LHYPERFINE = .TRUE.
 NGYROMAG = 10.7084 3.077

 #LASPH = .TRUE.

 ISPIN = 2

Change the values of NGYROMAG to your corresponding system. LASPH is turned off as, although it can be a contributing factor, it should be tested in each case. An extended description is given below.

Output

As usual, all output is written to the OUTCAR file. VASP writes three blocks of data. The first is for the Fermi contact coupling parameter:

 Fermi contact (isotropic) hyperfine coupling parameter (MHz)
 -------------------------------------------------------------
  ion      A_pw      A_1PS     A_1AE     A_1c      A_tot
 -------------------------------------------------------------
   1       ...       ...       ...       ...       ...
  ..       ...       ...       ...       ...       ...

 -------------------------------------------------------------

with an entry for each ion on the POSCAR file. A_pw, A_1PS, A_1AE, and A_1c are the plane wave, pseudo one-center, all-electron one-center, and one-center core contributions to the Fermi contact term, respectively. The total Fermi contact term is given by A_tot.

Important: We have chosen NOT to include the core contributions A_1c in A_tot. These are important to add when comapring to experiment where they can contribute a significant proportion to the hyperfine coupling constant (up to ~50 % for ¹³C ^[2]). If you want them to be included, you should add them by hand to A_tot:

{\displaystyle A_{tot + 1c} = A_{tot} + A_{1c} = (A_{pw} + A_{PS} + A_{AE}) + A_{1c} }

Core electronic contributions to the Fermi contact term are calculated in the frozen valence approximation as proposed by Yazyev et al.^[3].

The dipolar contributions are listed next:

 Dipolar hyperfine coupling parameters (MHz)
 ---------------------------------------------------------------------
  ion      A_xx      A_yy      A_zz      A_xy      A_xz      A_yz
 ---------------------------------------------------------------------
   1       ...       ...       ...       ...       ...       ...
  ..       ...       ...       ...       ...       ...       ...

 ---------------------------------------------------------------------

Again one line per ion in the POSCAR file.

The total hyperfine tensors are written as:

 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       ...       ...       ...         ...
  ..       ...       ...       ...         ...

 ----------------------------------------------------------------------

i.e., the tensors have been diagonalized and rearranged.

Mind: The Fermi contact term is strongly dominated by the all-electron one-center contribution A_1AE.

Unfortunately, this particular term is quite sensitive to the number and eigenenergy of the all-electron partial waves that make up the one-center basis set, i.e., to the particulars of the PAW dataset you are using. As a result, the Fermi contact term may strongly depend on the choice of PAW dataset.

Units

The Fermi contact term $A$ is measured in following units

${\displaystyle [A]= \left[\mu_0\right]\times \left[g_e \mu_e\right]\times \left[g_j \mu_j\right]\times \left[|\psi(0)|^2\right] = \frac{T^2m^3}{J}\times \frac{J}{T}\times \frac{MHz}{T}\times \frac{1}{m^3} = MHz }$

with $\mu_0=4\pi\times 10^{-7} T^2 m^3 J^{-1}$ , $g_e\mu_e=9.28476377\times 10^{-24} J T^{-1}, |\psi(0)|^2=10^{30}m^{-3}$ . NGYROMAG is given in units of MHz/T, see here for a table of different gyromagnetic ratios.

Advice

Choice of PAW potentials: The hyperfine coupling parameter can be sensitive to the specific PAW potential used, as different pseudopotentials include a varying number of electrons in the valence. It is important to match the all-electron (AE) wavefunction. GW pseudopotentials are often better at this than standard potentials.
The use of hybrid functionals can also improve the hyperfine coupling constants when compared to experiment ^[2].
We recommend using tightly converged settings:

PREC = Accurate
EDIFF = 1E-8

Additionally, we recommend performing convergence tests with respect to the plane-wave energy cutoff ENCUT and k-point mesh KPOINTS to ensure convergence has been achieved for your system.
It is possible that your system relaxes to a non-magnetic solution, causing the hyperfine splitting to disappear. If you think your system should be magnetic, you can enforce it using NUPDOWN, which will return the hyperfine splitting, cf. forum post: https://vasp.at/forum/viewtopic.php?t=16921. NUPDOWN will change the Total magnetic moment S= at the start of the hyperfine coupling section in the OUTCAR.

References

[bloechl:prb:2000-1] P. Bloechl, First-principles calculations of defects in oxygen-deficient silica exposed to hydrogen, Phys. Rev. B, 62, 6158 (2000).

[szasz:prb:2013-2] 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).

[yazyev:prb:2005-3] 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).

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