Zero field splitting

Zero field splitting describes various interactions of the energy levels of a molecule or ion resulting from the presence of more than one unpaired electron. The unpaired electrons mutually interact to give rise to two or more energy states. It is well known that degeneracy is lifted in the presence of a magnetic field, but zero field splitting occurs even in the absence of a magnetic field. ZFS is responsible for many effects in related to the magnetic properties of materials, as manifested in their electron spin resonance spectra and magnetism.

The classic case for ZFS is the spin triplet, i.e., the S=1 spin system. In the presence of a magnetic field, the levels with different values of magnetic spin quantum number (M_S=0,±1) are separated and the Zeeman splitting dictates their separation. In the absence of magnetic field, the 3 levels of the triplet are isoenergetic to the first order. However, when we incorporate the effect one electron has on the other, the energy of the sublevels of the triplet are separated out (zero field splitting). The separation of these sub levels manifest in peaks of the frozen solution EPR spectrum.

Quantum mechanical description

The correspondig Hamiltonian can be written as:

$\hat{\mathcal{H}}=D\left(S_z^2-\frac{1}{3}S(S+1)\right)+E(S_x^2-S_y^2)$

Where S is the total Spin quantum number, and $S_{x,y,z}$ are the spin matrices. The value of the ZFS parameter are usually defined via D and E parameters. D describes the axial component of the magnetic dipole-dipole interaction, and E the transversal component. Values of D have been obtained for a wide number of organic biradicals by EPR/ESR measurements. This value may be measured by other magnetometry techniques such as SQUID; however, EPR measurements provide more accurate data in most cases. This values can also be obtained with other techniques such as optically detected magnetic resonance (ODMR), with sensitivity down to a single molecule or defect in solids like Diamond (eg. N-V center) or Silicon Carbide.

Algebraic derivation

The start is the corresponding Hamiltonian $\hat{\mathcal{H}}_D=\mathbf{SDS}$. $\mathbf{D}$ describes the dipolar spin-spin interaction between two unpaired spins ($S_1$ and $S_2$). Where $S$ is the total spin $S=S_1+S_2$, and $\mathbf{D}$ being a symmetric and traceless (which it is when $\mathbf{D}$ arises from dipole-dipole interaction) matrix, which means it is diagonalizable.

$\mathbf{D}= \begin{pmatrix} D_{xx} & 0 & 0 \\ 0 & D_{yy} & 0 \\ 0 & 0 & D_{zz} \end{pmatrix}$

(1)

with $\mathbf{D}$ being traceless ($D_{xx}+D_{yy}+D_{zz}=0$). For simplicity $D_{j}$ is defined as $D_{jj}$. The Hamiltonian becomes:

$\hat{\mathcal{H}}_D=D_x S_x^2+D_y S_y^2+D_z S_z^2$

(2)

The key is to express $D_x S_x^2+D_y S_y^2$ as its mean value and a deviation $\Delta$

$D_x S_x^2+D_y S_y^2= \frac{D_x+D_y}{2}(S_x^2+S_y^2)+\Delta$

(3)

To find the value for the deviation $\Delta$ which is then by rearranging equation (3):

$\begin{align} \Delta & = \frac{D_x-D_y}{2}S_x^2+ \frac{D_y-D_x}{2}S_y^2\\ & = \frac{D_x-D_y}{2}(S_x^2-S_y^2) \end{align}$

(4)

By inserting (4) and (3) into (2) the result reads as:

$\begin{align} \hat{\mathcal{H}}_D & = \frac{D_x+D_y}{2}(S_x^2+S_y^2)+\frac{D_x-D_y}{2}(S_x^2-S_y^2)+D_zS_z^2 \\ & = \frac{D_x+D_y}{2}(S_x^2+S_y^2+S_z^2-S_z^2)+\frac{D_x-D_y}{2}(S_x^2-S_y^2)+D_zS_z^2 \end{align}$

(5)

Note, that in the second line in (5) $S_z^2-S_z^2$ was added. By doing so $S_x^2+S_y^2+S_z^2=S(S+1)$ can be further used. By using the fact, that $\mathbf{D}$ is traceless ($\frac{1}{2}D_x+\frac{1}{2}D_y=-\frac{1}{2}D_z$) equation (5) simplifies to:

$\begin{align} \hat{\mathcal{H}}_D & = -\frac{D_z}{2}S(S+1)+\frac{1}{2}D_z S_z^2+\frac{D_x-D_y}{2}(S_x^2-S_y^2)+D_zS_z^2 \\ & =-\frac{D_z}{2}S(S+1)+\frac{3}{2}D_z S_z^2+\frac{D_x-D_y}{2}(S_x^2-S_y^2) \\ & =\frac{3}{2}D_z \left( S_z^2-\frac{S(S+1)}{3} \right)+\frac{D_x-D_y}{2}(S_x^2-S_y^2) \end{align}$

(6)

By defining D and E parameters equation (6) becomes to:

$\hat{\mathcal{H}}_D =D\left(S_z^2-\frac{1}{3}S(S+1)\right)+E(S_x^2-S_y^2)$

(7)

with $D=\frac{3}{2}D_z$ and $E=\frac{1}{2}\left(D_x-D_y\right)$ (measurable) zero field splitting values.

External links

• Description of the origins of Zero Field Splitting

References

Bibliography

• Atherton, N.M. (1993). Principles of electron spin resonance. Ellis Horwood PTR Prentice Hall. doi:10.1016/0307-4412(95)90208-2. ISBN 0-137-21762-5.

Principles of electron spin resonance: By N M Atherton. pp 585. Ellis Horwood PTR Prentice Hall. 1993 ISBN 0-137-21762-5

• Christle, David J.; et, al (2015). "Isolated electron spins in silicon carbide with millisecond coherence times" (HTML). Nature Materials 14 (6): 160–163. doi:10.1038/nmat4144. Retrieved June 27, 2014.

• Widmann, Matthias; et, al (2015). "Coherent control of single spins in silicon carbide at room temperature" (HTML). Nature Materials 14 (6): 164–168. doi:10.1038/nmat4145. Retrieved June 29, 2014.