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) [1]:
- [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. [2].
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. |
Input
An example INCAR is presented below used for a Nitrogen-vacancy (NV) center in diamond:
PREC = Accurate ENCUT = 400 EDIFF = 1E-6 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 commented out as for crystals, it is not generally important; however, for molecules, it is very important.
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 comparing to experiment where they can contribute a significant proportion to the hyperfine coupling constant (up to ~50 % for 13C [1]). 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.
Advice
PAW pseudopotentials
- 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 (i.e. _sv, _pv). It is important to match the all-electron (AE) wavefunction.
- GW pseudopotentials (i.e. _GW) often offer a better description than standard potentials.
Core contributions and hybrid functionals
- Make sure to include core contributions when reporting Fermi contact terms [3]. These are very important and their inclusion improves comparison to experiment [1].
- The use of hybrid functionals can also improve the hyperfine coupling constants when compared to experiment. E.g., for defects in silicon, HSE06 localizes the defect relative to PBE, significantly improving the description [1].
Input parameters
- We recommend using tightly converged settings:
PREC = Accurate EDIFF = 1E-6 # Note that some systems might require tighter settings, e.g. 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.
- Convergence is generally quick with respect to energy cutoff.
- Convergence can be very dependent on k-point mesh density. In some cases, e.g. if your cell is too small, then increasing the k-point mesh can result in convergence to a non-magnetic solution, eliminating the hyperfine coupling.
- Test your system with LASPH = .TRUE. and .FALSE. In some cases, non-spherical contributions may be important.
Converging to non-magnetic states
- It is possible that your system relaxes to a non-magnetic solution, causing the hyperfine splitting to disappear (i.e. all zeros). 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.
Important: For some cells, the total magnetic moment S can be very small (grep " mag=" OSZICAR), near zero. In the above equations, the isotropic and anisotropic components of the hyperfine coupling parameter (AIiso and AIani) are calculated by dividing through by S (cf. ⟨Sz⟩). To avoid division by zero, S is reset to 1 when S < 10-3. Total magnetic moment S= is changed, changing the hyperfine coupling constants, too. These hyperfine coupling constants are therefore not meaningful as far as we know. In future versions of the code, there will be a warning message stating that S has been reset and printing the correct total magnetic moment.
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Related tags and articles
References
- ↑ a b c d 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).
- ↑ P. Bloechl, First-principles calculations of defects in oxygen-deficient silica exposed to hydrogen, Phys. Rev. B, 62, 6158 (2000).
- ↑ a b 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).