Zero field splitting

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Zero field splitting (ZFS) describes various interactions of the energy levels of a

The corresponding Hamiltonian can be written as:

${\displaystyle {\hat {\mathcal {H}}}=D\left(S_{z}^{2}-{\frac {1}{3}}S(S+1)\right)+E(S_{x}^{2}-S_{y}^{2})}$

Where S is the total spin quantum number, and ${\displaystyle S_{x,y,z}}$ 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

.

Algebraic derivation

The start is the corresponding Hamiltonian ${\displaystyle {\hat {\mathcal {H}}}_{D}=\mathbf {SDS} }$. ${\displaystyle \mathbf {D} }$ describes the dipolar spin-spin interaction between two unpaired spins (${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$). Where ${\displaystyle S}$ is the total spin ${\displaystyle S=S_{1}+S_{2}}$, and ${\displaystyle \mathbf {D} }$ being a symmetric and traceless (which it is when ${\displaystyle \mathbf {D} }$ arises from dipole-dipole interaction) matrix, which means it is diagonalizable.

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

(1)

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

${\displaystyle {\hat {\mathcal {H}}}_{D}=D_{x}S_{x}^{2}+D_{y}S_{y}^{2}+D_{z}S_{z}^{2}}$

(2)

The key is to express ${\displaystyle D_{x}S_{x}^{2}+D_{y}S_{y}^{2}}$ as its mean value and a deviation ${\displaystyle \Delta }$

${\displaystyle D_{x}S_{x}^{2}+D_{y}S_{y}^{2}={\frac {D_{x}+D_{y}}{2}}(S_{x}^{2}+S_{y}^{2})+\Delta }$

(3)

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

${\displaystyle {\begin{aligned}\Delta &={\frac {D_{x}-D_{y}}{2}}S_{x}^{2}+{\frac {D_{y}-D_{x}}{2}}S_{y}^{2}\\&={\frac {D_{x}-D_{y}}{2}}(S_{x}^{2}-S_{y}^{2})\end{aligned}}}$

(4)

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

${\displaystyle {\begin{aligned}{\hat {\mathcal {H}}}_{D}&={\frac {D_{x}+D_{y}}{2}}(S_{x}^{2}+S_{y}^{2})+{\frac {D_{x}-D_{y}}{2}}(S_{x}^{2}-S_{y}^{2})+D_{z}S_{z}^{2}\\&={\frac {D_{x}+D_{y}}{2}}(S_{x}^{2}+S_{y}^{2}+S_{z}^{2}-S_{z}^{2})+{\frac {D_{x}-D_{y}}{2}}(S_{x}^{2}-S_{y}^{2})+D_{z}S_{z}^{2}\end{aligned}}}$

(5)

Note, that in the second line in (5) ${\displaystyle S_{z}^{2}-S_{z}^{2}}$ was added. By doing so ${\displaystyle S_{x}^{2}+S_{y}^{2}+S_{z}^{2}=S(S+1)}$ can be further used. By using the fact, that ${\displaystyle \mathbf {D} }$ is traceless (${\displaystyle {\frac {1}{2}}D_{x}+{\frac {1}{2}}D_{y}=-{\frac {1}{2}}D_{z}}$) equation (5) simplifies to:

${\displaystyle {\begin{aligned}{\hat {\mathcal {H}}}_{D}&=-{\frac {D_{z}}{2}}S(S+1)+{\frac {1}{2}}D_{z}S_{z}^{2}+{\frac {D_{x}-D_{y}}{2}}(S_{x}^{2}-S_{y}^{2})+D_{z}S_{z}^{2}\\&=-{\frac {D_{z}}{2}}S(S+1)+{\frac {3}{2}}D_{z}S_{z}^{2}+{\frac {D_{x}-D_{y}}{2}}(S_{x}^{2}-S_{y}^{2})\\&={\frac {3}{2}}D_{z}\left(S_{z}^{2}-{\frac {S(S+1)}{3}}\right)+{\frac {D_{x}-D_{y}}{2}}(S_{x}^{2}-S_{y}^{2})\end{aligned}}}$

(6)

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

${\displaystyle {\hat {\mathcal {H}}}_{D}=D\left(S_{z}^{2}-{\frac {1}{3}}S(S+1)\right)+E(S_{x}^{2}-S_{y}^{2})}$

(7)

with ${\displaystyle D={\frac {3}{2}}D_{z}}$ and ${\displaystyle E={\frac {1}{2}}\left(D_{x}-D_{y}\right)}$ (measurable) zero field splitting values.

References