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.

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

As usual, all output is written to the OUTCAR file. VASP writes three blocks of data, that look something like:

 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. Beware: for the moment we have chosen NOT to include the core contributions A_1c in A_tot. If you so want, you should add them by hand to A_tot. Core electronic contributions to the Fermi contact term are calculated in the frozen valence approximation as proposed by Yazyev et al.^[2]

The dipolar constributions 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.

N.B.: 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.

References

[bloechl:prb:00-1] P. E. Blöchl, Phys. Rev. B 62, 6158 (2000).

[yazyev:prb:05-2] O. V. Yazyev, I. Tavernelli, L. Helm, and U. R. Röthlisberger, Phys. Rev. B 71, 115110 (2006).

[1]

[2]

Units

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References