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 AI describes the interaction between a nuclear spin SI (located at site RI) and the electronic spin distribution Se (in most cases associated with a paramagnetic defect state):
- [math]\displaystyle{ E=\sum_{ij} S^e_i A^I_{ij} S^I_j }[/math]
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
- [math]\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} }[/math]
where ρs is the spin density, μ0 is the magnetic susceptibility of free space, γe the electron gyromagnetic ratio, γI the nuclear gyromagnetic ratio of the nucleus at RI, and [math]\displaystyle{ \langle S_z \rangle }[/math] the expectation value of the z-component of the total electronic spin.
δT(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
- [math]\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} }[/math]
In the equations above r=|r|, ri the i-th component of r, and r is taken relative to the position of the nucleus RI.
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. |
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. Apw, A1PS, A1AE, and A1c 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 Atot.
Important: We have chosen NOT to include the core contributions A1c in Atot. These are important to add when comapring to experiment where they can contribute up to ~50 % of the hyperfine coupling constant [2]. If you want them to be included, you should add them by hand to Atot:
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 A1AE.
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 [math]\displaystyle{ A }[/math] is measured in following units
[math]\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 }[/math]
with [math]\displaystyle{ \mu_0=4\pi\times 10^{-7} T^2 m^3 J^{-1} }[/math], [math]\displaystyle{ g_e\mu_e=9.28476377\times 10^{-24} J T^{-1}, |\psi(0)|^2=10^{30}m^{-3} }[/math]. 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
- Additional, 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.
Related tags and articles
References
- ↑ P. Bloechl, First-principles calculations of defects in oxygen-deficient silica exposed to hydrogen, Phys. Rev. B, 62, 6158 (2000).
- ↑ a b 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).