The nitrogen hyperfine interaction

The hyperfine parameters as given in table 7-1 are non-axial; that is to say A1 x

A2 xA3. This may be due to a number of reasons, including core spin-polarisation

and extended interactions with higher-order orbitals [36], or the symmetry of the defect in question.

A simple treatment suggests that each nitrogen atom possesses a lone pair pointing into the vacancy, and that by Pauli exclusion further electrons must reside in higher energy states: therefore 50% of the unpaired electron probability density is

assumed to be localised at a point along each `1 1 1e “bond” between each of the

carbon atoms and the vacancy. In the simple case (ignoring spin-orbit effects), the hyperfine interaction can be described as the sum of an isotropic component

a arising from non-zero spin density at the nucleus, and an anisotropic dipolar

component b. The isotropic component of the hyperfine interaction at each point

is then a ˆA1A2A3~6 2.01 MHz. In its principal axis frame, the matrix

describing a dipolar interaction is diagonal with values b,b,2b. In considering

the dipolar component, a dipolar matrix is constructed at each of the assumed spin density locations, and then transformed into the crystal axes. The matrices may be summed to give a macroscopic dipolar interaction matrix, and compared against the experimentally measured values.

Using the approach described above, a calculation for the ˆNVN defect was

performed. The interaction was constrained to the ˜1 1 0 planes containing one

each of the nitrogen and carbon atoms, and the vacancy. The only free parameters

wereθ, the in-plane angle between the interaction vector and the `1 1 1e direction

connecting the nitrogen and vacancy, and b, the strength of the interaction in

MHz. θ and b were varied to minimise the difference between the observed values

and the calculated values. The values which give the best fit to experiment are given in table 7-2.

The best fit is achieved at an angle ofθ 35.26°, and an interaction strength of b

0.46 MHz. This corresponds to 50 % of the electron spin density being localised

at each of the carbon atoms, with the hyperfine interaction for each location being

along the `1 1 0e direction between the carbon and nitrogen atoms — consistent

with the initial assumptions of the electron localisation of the carbon atoms. The resulting macroscopic interaction is a good fit to experiment; a comparison of eigenvalues and eigenvectors is given in table 7-2b.

For high-symmetry hyperfine interactions, the relative signs of the principal hy- perfine values are ambiguous; the construction of the macroscopic interaction in

120°

1 1 1

(a) The localisation of the unpaired electron probability density along the “bonds” between each carbon and the vacancy. The planes which in whichθ

is defined (see text and below figure) are shown as dotted lines.

θ

1 1 1

(b)The angleθis defined as the angle between the`1 1 1edirection connecting the vacancy and nitrogen sites, and the localised unpaired electron probability density. The angle is confined to the ˆ1 1 0plane which contains all three lattice sites.

Figure 7-8 The geometry used for the dipolar calculation described in the text. The unpaired electron probability density is localised at

a point along the `1 1 1e “bond” between each of the carbon atoms

and the vacancy (orange dot): at each of these two points a dipolar interaction matrix is constructed and transformed into the crystal axes. The results are summed to give a macroscopic interaction.

this way allows the same sign to be attributed to all three principal values. The local spin-DFT calculations performed by Goss [37] (using the methods described in [38]) also give a similar polarity to all values, the absolute sign of which could be confirmed experimentally by ENDOR measurements.

In order to check the calculated values are reasonable, the expected interaction magnitude may be calculated. For a simple dipolar interaction, the interaction strength is given by b µ0 4π geµBgnµn h c 1 r3h ,

whereris the distance between the unpaired electron and the nucleus, assumed to

be a point dipole. For a 15_{N nucleus at a distance r 2.52A (next nearest neigh-}

b (MHz) Eigenvalues (MHz) Eigenvectors

0.46 0.58,0.12,0.46 1 1 2, 1 1 0, 1 1 1

(a) Orientation and interaction strength of the macroscopic dipolar matrix for each nitrogen atom. Constructed using an interaction strength of b at each carbon atom and summed in the crystal axes.

Eigenvalues (MHz) Eigenvectors

Experiment 3.47ˆ2, 4.51ˆ2, 4.09ˆ2 3.5° from 1 1 2, 3.5° from 1 1 1, 1 1 0

Simulation 3.44, 4.48, 4.14 1 1 2, 1 1 1, 1 1 0

DFT[37] 2.6, 3.7, 3.2 3.8° from 1 1 2, 3.8° from 1 1 1, 1 1 0

(b) Comparison of the experimental and calculated hyperfine values. Cal- culated values are as those given above with the isotropic component 22.01 MHz added.

Table 7-2 Results of calculations designed to minimise the difference

between the calculated and observed hyperfine values forˆNVN. See

text for details.

by least squares fit). Given the crudeness of the model employed, the agreement with experiment is remarkably good. No geometric relaxation was included in this model, and the electron spin density was taken to be localised at two points only. For a more precise calculation, information about shape and amplitude of the electron wavefunction is required in addition to knowledge about the minimum energy geometrical configuration.