Editing Spintronics (section)

== Mathematical Equations in Spintronics ==

In Spintronics, several mathematical equations are fundamental to understanding the behavior of spin currents, magnetoresistance, and other key phenomena. This section provides an overview of these important equations.

=== [[Spin Current Density]] ===
The spin current density, which represents the flow of electron spins in a material, can be expressed as:

<math>
\mathbf{J}_s = \frac{\hbar}{2e} \mathbf{J}_c \cdot \mathbf{P}
</math>

where:
* <math>\mathbf{J}_s</math> is the spin current density,
* <math>\hbar</math> is the reduced Planck constant,
* <math>e</math> is the elementary charge,
* <math>\mathbf{J}_c</math> is the charge current density,
* <math>\mathbf{P}</math> is the spin polarization vector.

=== [[Giant Magnetoresistance (GMR)]] ===
The giant magnetoresistance effect, which is the change in electrical resistance due to the alignment of magnetic moments, can be described by:

<math>
R = R_0 \left(1 + \Delta R \cdot \cos\theta\right)
</math>

where:
* <math>R</math> is the resistance,
* <math>R_0</math> is the base resistance,
* <math>\Delta R</math> is the change in resistance,
* <math>\theta</math> is the angle between the magnetizations of adjacent layers.

=== [[Spin Diffusion Length]] ===
The spin diffusion length, which characterizes how far spin information can travel in a material before it loses coherence, is given by:

<math>
\lambda_s = \sqrt{D_s \tau_s}
</math>

where:
* <math>\lambda_s</math> is the spin diffusion length,
* <math>D_s</math> is the spin diffusion coefficient,
* <math>\tau_s</math> is the spin relaxation time.

=== [[Spin Hall Effect]] ===
The spin Hall effect, where a spin current is generated perpendicular to an applied charge current, can be expressed as:

<math>
\mathbf{J}_s = \theta_{SH} \left(\mathbf{J}_c \times \mathbf{e}_z\right)
</math>

where:
* <math>\mathbf{J}_s</math> is the spin current density,
* <math>\theta_{SH}</math> is the spin Hall angle,
* <math>\mathbf{J}_c</math> is the charge current density,
* <math>\mathbf{e}_z</math> is the unit vector in the direction perpendicular to the current flow.

=== [[Spin Torque]] ===
The spin transfer torque, which describes the transfer of spin angular momentum from a spin current to the magnetization of a material, is given by:

<math>
\mathbf{\tau}_{ST} = \frac{\hbar}{2e} \frac{\mathbf{J}_s \times \mathbf{M}}{M_s}
</math>

where:
* <math>\mathbf{\tau}_{ST}</math> is the spin torque,
* <math>\hbar</math> is the reduced Planck constant,
* <math>e</math> is the elementary charge,
* <math>\mathbf{J}_s</math> is the spin current density,
* <math>\mathbf{M}</math> is the magnetization vector,
* <math>M_s</math> is the saturation magnetization.

=== [[Spin Polarization]] ===
The degree of spin polarization in a material, which measures the imbalance between spin-up and spin-down electrons, is expressed as:

<math>
P = \frac{n_{\uparrow} - n_{\downarrow}}{n_{\uparrow} + n_{\downarrow}}
</math>

where:
* <math>P</math> is the spin polarization,
* <math>n_{\uparrow}</math> is the density of spin-up electrons,
* <math>n_{\downarrow}</math> is the density of spin-down electrons.

=== [[Rashba Spin-Orbit Interaction]] ===
The Rashba spin-orbit interaction, which occurs in systems lacking structural inversion symmetry, is described by:

<math>
H_{R} = \alpha_R \left(\mathbf{k} \times \mathbf{\sigma}\right) \cdot \mathbf{e}_z
</math>

where:
* <math>H_{R}</math> is the Rashba Hamiltonian,
* <math>\alpha_R</math> is the Rashba coefficient,
* <math>\mathbf{k}</math> is the wave vector,
* <math>\mathbf{\sigma}</math> is the vector of Pauli matrices,
* <math>\mathbf{e}_z</math> is the unit vector perpendicular to the plane of the system.

<sub>''Caption:'' These equations represent some of the fundamental mathematical concepts in spintronics, describing spin current density, magnetoresistance, spin diffusion, and related phenomena.''</sub>