- A fundamental model in magnetism used to describe the energy of interacting spins.
- The model treats each magnetic moment as a classical vector.
$$ \begin{aligned} \mathcal{H} = & -\sum_{\langle i,j \rangle} \Bigl( J_{ij},\mathbf{S}_i\cdot\mathbf{S}j + \mathbf{D}{ij}\cdot (\mathbf{S}_i\times\mathbf{S}j) \Bigr)\ &-\sum{i} K (\mathbf{S}i\cdot \mathbf{n})^2 -\sum_i \mu_i, \mathbf{B}\cdot\mathbf{S}i\ &+ \frac{\mu_0}{4\pi} \sum{i<j} \frac{\mu_i,\mu_j}{r{ij}^3} \left[\mathbf{S}_i\cdot\mathbf{S}_j - 3,(\mathbf{S}i\cdot \hat{r}{ij})(\mathbf{S}j\cdot \hat{r}{ij})\right]\ &+ \text{other terms} \end{aligned} $$
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Cheap to evaluate. Simulations with
$10^6$ spins over nanoseconds are possible. -
Coarse-graining the Heisenberg Hamiltonian is the basis of micromagnetics
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Exchange Interaction:
$-\sum_{\langle i,j \rangle} J_{ij}, \mathbf{S}_i\cdot\mathbf{S}_j$ -
$J_{ij}$ : Exchange coupling constant between spins at sites$i$ and$j$ . -
$J_{ij} > 0$ favors ferromagnetic alignment, while$J_{ij} < 0$ favors antiferromagnetic alignment. -
$\mathbf{S}_i$ and$\mathbf{S}_j$ : Spin vectors (unit vectors) at sites$i$ and$j$
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Dzyaloshinskii–Moriya Interaction (DMI):
$-\sum_{\langle i,j \rangle} \mathbf{D}_{ij}\cdot (\mathbf{S}_i\times\mathbf{S}_j)$ -
$\mathbf{D}_{ij}$ : DMI vector for the spin pair at sites$i$ and$j$ . - Favors chiral (twisted or non-collinear) spin configurations.
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Anisotropy:
$-\sum_{i} K, (\mathbf{S}_i\cdot \mathbf{n})^2$ -
$K$ : Anisotropy constant. -
$\mathbf{n}$ : Preferred (easy) axis direction. - Encourages spins to align along the direction of
$\mathbf{n}$ .
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Zeeman Term:
$-\sum_i \mu_i, \mathbf{B}\cdot\mathbf{S}_i$ -
$\mu_i$ : Magnetic moment at site$i$ . -
$\mathbf{B}$ : External magnetic field. - Represents the interaction of the spins with an applied magnetic field.
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Dipole–Dipole Interaction (DDI): $\frac{\mu_0}{4\pi}\sum_{i<j} \frac{\mu_i,\mu_j}{r_{ij}^3}\left[\mathbf{S}_i\cdot\mathbf{S}_j - 3, (\mathbf{S}i\cdot\hat{r}{ij})(\mathbf{S}j\cdot\hat{r}{ij})\right]$
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$\mu_0$ : Permeability of free space. -
$\mu_i$ and$\mu_j$ : Magnetic moments at sites$i$ and$j$ . -$r_{ij}$ : - Distance between spins at sites
$i$ and$j$ . -
$\hat{r}_{ij}$ : Unit vector pointing from site$i$ to site$j$ . - Accounts for the long-range dipolar interaction between spins.
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The LLG equation governs the time evolution of magnetic moments (spins) in a material under the influence of effective magnetic fields. It can be written as $$ \frac{d\mathbf{S}}{dt} = -\frac{\gamma}{1+\alpha^2},\mathbf{S}\times\Bigl( \mathbf{H}{\mathrm{eff}} + \boldsymbol{\zeta}(t) \Bigr) -\frac{\gamma\alpha}{1+\alpha^2},\mathbf{S}\times\Bigl[\mathbf{S}\times\Bigl( \mathbf{H}{\mathrm{eff}} + \boldsymbol{\zeta}(t) \Bigr)\Bigr], $$
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$\mathbf{S}$ : Unit vector representing the spin. -
$\mathbf{H}_{\mathrm{eff}}$ : Effective magnetic field (includes contributions from exchange, anisotropy, Zeeman, etc.). -
$\gamma$ : Gyromagnetic ratio. -
$\alpha$ : Gilbert damping parameter. -
$\boldsymbol{\zeta}(t)$ : Stochastic thermal field representing thermal fluctuations. -
The Langevin noise-term
$\zeta(t)$ can be used to model the time-evolution of a spin system in the canonical (NVT) ensemble. It is a Gaussian noise term with zero mean and the correlation function $$ \langle \zeta_i(t) , \zeta_j(t') \rangle = \frac{2\alpha, k_B T}{\gamma \mu_s} , \delta_{ij}, \delta(t-t'), $$
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The exercised should be small and self-contained. They should be represented in Jupyter notebooks. TODO.