Spin-orbit coupling
Spin-orbit coupling
Spin-orbit coupling (SOC) is a relativistic effect originating from the relative motion between the valence electron and the nucleus. If an electron is in a fixed inertial frame so that its velocity is zero ($\mathbf{v}_{\rm e} = \mathbf{0}$), it sees the nucleus orbiting around it by $\mathbf{v}$ creating an electric field $\mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r})$. Here, $V(\mathbf{r})$ is the central potential of the nucleus, and it generally includes the screening effect from core electrons. The magnetic field that the electron experiences is expressed as:
\[\begin{equation} \mathbf{B}_{\rm eff} = -\frac{\mathbf{v}}{c}\times \mathbf{E} \end{equation}\]If SOC is absent, this magnetic field also does not exit. Then the spin has the full rotational symmetry and the energy becomes the same in every spin direction. Thus, once the $z$-axis is set to be the referential axis, $S_{z}$ becomes the good quantum number, and the eigenstate is expressed by the spin state (up or down) and spatial wave function $\psi_{nk}(\mathbf{r})$.
However, if SOC is present, the eigenstate is given by the linear combination of spin and orbital degrees of freedom. The electron will feel the effective magnetic field generated by the nuclues. This implies that the spin degree of freedom $\mathbf{S}$ is coupled to the orbital angular momentum $\mathbf{L}$.
Note. In crystal, the symmetry of the orbital angular momentum lowers and it follows the crystal’s one. Since the spin degree of freedom is coupled to orbital one, the spin favors a specific direction governed by the crystal symmetry. This leads to the magnetic anisotropy.
Spin-orbit coupling in the lattice system
Let the basis $\ket{i\alpha\sigma} = \ket{\psi_{i\alpha}}\otimes\ket{\sigma}$. Here, $i$, $\alpha$, and $\sigma$ denote site, orbita, and spin indices. The matrix element of the Hamiltonian with respect to spin-orbit coupling is defined as:
\[\begin{equation} h^{i\alpha\sigma,j\beta\sigma'}_{\rm SOC} = \braket{i\alpha\sigma|H_{\rm SOC}|j\beta\sigma'} \end{equation}\]where $H_{\rm SOC}$ is expressed as follows:
\[\begin{equation} H_{\rm SOC} = \frac{\hbar}{4m^2c^2}\boldsymbol{\sigma}\cdot[\nabla V(\mathbf{r})\times \mathbf{p}] \end{equation}\]The Pauli spin matrix $\boldsymbol{\sigma}$ acts only on the spin subspace, $\nabla V(\mathbf{r})\times\mathbf{p}$ acts on the orbital subspace. These two operators can be separated by:
\[\begin{equation} h^{i\alpha\sigma,j\beta\sigma'}_{\rm SOC} = \sum_{\mu = x,y,z} O^{\mu}_{i\alpha,j\beta} \sigma^{\mu}_{\sigma\sigma'} \end{equation}\]where $O^{\mu}_{i\alpha,j\beta}$ is given by:
\[\begin{align} O^{\mu}_{i\alpha,j\beta} &= \frac{\hbar}{4m^2c^2}\int d\mathbf{r}~\phi^{*}_{i\alpha}(\mathbf{r}) [\nabla V(\mathbf{r})\times\mathbf{p}]_{\mu}\phi_{j\beta}(\mathbf{r}) \nonumber \\ &= \frac{\hbar}{4m^2c^2}\int d\mathbf{r}~\phi^{*}_{i\alpha}(\mathbf{r})\left[\sum_{\nu,\rho}\varepsilon_{\mu\nu\rho} (\partial_{\nu}V)p_{\rho}\right]\phi_{j\beta}(\mathbf{r}) \nonumber \\ &= \frac{\hbar}{4m^2c^2}\int d\mathbf{r}~\phi^{*}_{i\alpha}(\mathbf{r})\left[\sum_{\nu,\rho}\varepsilon_{\mu\nu\rho} (\partial_{\nu}V)(-i\hbar\partial_{\rho})\right]\phi_{j\beta}(\mathbf{r}) \nonumber \\ &= -i \frac{\hbar^{2}}{4m^2c^2}\sum_{\nu,\rho}\varepsilon_{\mu\nu\rho}\int d\mathbf{r}~\phi^{*}_{i\alpha}(\mathbf{r}) \partial_{\nu}V\partial_{\rho}\phi_{j\beta}(\mathbf{r}) \end{align}\]If $\phi_{i\alpha}$ and $\phi_{j\beta}$ are real basis, one can simply rewrite it using $\lambda^{\mu}_{i\alpha,j\beta}$:
\[\begin{equation} \lambda^{\mu}_{i\alpha,j\beta} = -\frac{\hbar^{2}}{4m^2c^2}\sum_{\nu,\rho}\varepsilon_{\mu\nu\rho}\int d\mathbf{r}~\phi^{*}_{i\alpha}(\mathbf{r}) \partial_{\nu}V\partial_{\rho}\phi_{j\beta}(\mathbf{r}) \end{equation}\]Finally, $h^{i\alpha\sigma,j\beta\sigma’}_{\rm SOC}$ is written as:
\[\begin{align} h^{i\alpha\sigma,j\beta\sigma'}_{\rm SOC} &= i\sum_{\mu = x,y,z} \lambda^{\mu}_{i\alpha,j\beta}\sigma^{\mu}_{\sigma\sigma'} \nonumber \\ &= i\boldsymbol{\lambda}_{i\alpha,j\beta}\cdot\boldsymbol{\sigma}_{\sigma\sigma'} \end{align}\]If the real orbital basis is assumed in the lattice model, the spin-orbit coupling has the imaginary factor $i$ that can be seen in many papers and textbooks. Now the import point is that the symmetry of the lattice is in $\boldsymbol{\lambda}_{i\alpha,j\beta}$ (because of $V(\mathbf{r})$).
Spin-orbit coupling without inversion symmetry
Assume the two-dimensional system without inversion symmetry along the $z$-axis. One has an interest in how the confinement potential affects the spin-orbit coupling. The change in the potential is given by $\nabla V \simeq (\partial_{z}V)\hat{z}$ and the spin-orbit coupling Hamiltonian then reads:
\[\begin{align} H_{\rm SOC} &= \frac{\hbar}{4m^2c^2}\boldsymbol{\sigma}\cdot[\nabla V(\mathbf{r})\times \mathbf{p}] \nonumber \\ &= \frac{\hbar}{4m^2c^2}\boldsymbol{\sigma}\cdot(\partial_{z} V)[p_{x}\hat{y}-p_{y}\hat{x}] \nonumber \\ &= \frac{\hbar}{4m^2c^2} (\partial_{z} V)[\sigma_y p_{x}-\sigma_{x}p_{y}] \end{align}\]The matrix element $h^{i\alpha,j\beta}_{\rm SOC}$ for this term is given by:
\[\begin{align} h^{i,i+\hat{x}}_{\rm SOC} &= i\alpha\sigma_{y} \\ h^{i,i+\hat{y}}_{\rm SOC} &= -i\alpha\sigma_{x} \end{align}\]where one uses the rule that $p_{x}$ survices for the $+x$-direction hopping and $p_{y}$ survices for the $+y$-direction hopping. In addition, $\alpha$ is written as:
\[\begin{equation} \alpha = -\frac{\hbar}{4m^2c^2}(\partial_{z}V) \end{equation}\]This Hamiltonian in the second quantization form is given by:
\[\begin{equation} \sum_{i}\left[c^{\dagger}_{i}(i\alpha\sigma_{y})c_{i+\hat{x}}+c^{\dagger}_{i}(-i\alpha\sigma_{x})c_{i+\hat{y}} \right] + \rm h.c. \end{equation}\]The Fourier transformation leads to:
\[\begin{equation} H_{R} = 2\alpha[\sin(k_{x}a)\sigma_{x} - \sin(k_{y}a)\sigma_{y}] \end{equation}\]This Hamiltonian is known as the Rashba Hamiltonian, and can be approximated in a small $k$ as $2\alpha a (k_{x} \sigma_{x} - k_{y} \sigma_{y})$.
References
- Sakurai, J. J. & Tuan, S. F. Modern Quantum Mechanics. (1985). p. 305
- Ashcroft, N. W. & Mermin, N. D. Solid State Physics. (2011). p. 186