In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is an interaction of a particle's spin with its motion. The first and best known example of this is that spin–orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and the magnetic field generated by the electron's orbit around the nucleus. This is detectable as a splitting of spectral lines, which can be thought of as a Zeeman Effect due to the internal field. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy.
Contents
- Spinorbit interaction in atomic energy levels
- Energy of a magnetic moment
- Magnetic field
- Magnetic moment of the electron
- Larmor interaction energy
- Thomas interaction energy
- Total interaction energy
- Evaluating the energy shift
- Final energy shift
- Spinorbit interaction in solids
- Examples of effective Hamiltonians
- Electron spin in inhomogeneous magnetic field
- Textbooks
- References
Spin–orbit interaction in atomic energy levels
This section presents a relatively simple and quantitative description of the spin–orbit interaction for an electron bound to an atom, up to first order in perturbation theory, using some semiclassical electrodynamics and non-relativistic quantum mechanics. This gives results that agree reasonably well with observations. A more rigorous derivation of the same result would start with the Dirac equation, and achieving a more precise result would involve calculating small corrections from quantum electrodynamics.
Energy of a magnetic moment
The energy of a magnetic moment in a magnetic field is given by:
where μ is the magnetic moment of the particle and B is the magnetic field it experiences.
Magnetic field
We shall deal with the magnetic field first. Although in the rest frame of the nucleus, there is no magnetic field acting on the electron, there is one in the rest frame of the electron (see Classical electromagnetism and special relativity). Ignoring for now that this frame is not inertial, in SI units we end up with the equation
where v is the velocity of the electron and E the electric field it travels through. Here weak relativistic is assumed that Lorentz factor
Next, we express the electric field as the gradient of the electric potential
where
It is important to note at this point that B is a positive number multiplied by L, meaning that the magnetic field is parallel to the orbital angular momentum of the particle, which is itself perpendicular to the particle's velocity.
Magnetic moment of the electron
The magnetic moment of the electron is
where
The spin–orbit potential consists of two parts. The Larmor part is connected to the interaction of the magnetic moment of the electron with the magnetic field of the nucleus in the co-moving frame of the electron. The second contribution is related to Thomas precession.
Larmor interaction energy
The Larmor interaction energy is
Substituting in this equation expressions for the magnetic moment and the magnetic field, one gets
Now, we have to take into account Thomas precession correction for the electron's curved trajectory.
Thomas interaction energy
In 1926 Llewellyn Thomas relativistically recomputed the doublet separation in the fine structure of the atom. Thomas precession rate,
where
To the first order in
Total interaction energy
The total spin–orbit potential in an external electrostatic potential takes the form
The net effect of Thomas precession is the reduction of the Larmor interaction energy by factor 1/2 which came to be known as the Thomas half.
Evaluating the energy shift
Thanks to all the above approximations, we can now evaluate the detailed energy shift in this model. In particular, we wish to find a basis that diagonalizes both H0 (the non-perturbed Hamiltonian) and ΔH. To find out what basis this is, we first define the total angular momentum operator
Taking the dot product of this with itself, we get
(since L and S commute), and therefore
It can be shown that the five operators H0, J2, L2, S2, and Jz all commute with each other and with ΔH. Therefore, the basis we were looking for is the simultaneous eigenbasis of these five operators (i.e., the basis where all five are diagonal). Elements of this basis have the five quantum numbers: n (the "principal quantum number") j (the "total angular momentum quantum number"), l (the "orbital angular momentum quantum number"), s (the "spin quantum number"), and jz (the "z-component of total angular momentum").
To evaluate the energies, we note that
for hydrogenic wavefunctions (here
Final energy shift
We can now say
where
Spin–orbit interaction in solids
A crystalline solid (semiconductor, metal etc.) is characterized by its band structure. While on the overall scale (including the core levels) the spin–orbit interaction is still a small perturbation, it may play a relatively more important role if we zoom in to bands close to the Fermi level (
In crystalline solid contained paramagnetic ions e.g. ions with unclosed d or f atomic subshell, localized electronic states exist. In this case, atomic-like electronic levels structure is shaped by intrinsic magnetic spin-orbit interactions and interactions with crystalline electric fields. Such structure is named the fine electronic structure. For rare-earth ions the spin-orbit interactions are much stronger than the CEF interactions. The strong spin-orbit coupling makes J become a relatively good quantum number, because the first excited multiplet is at least ~130 meV (1500 K) above the primary multiplet. The result is that filling it at room temperature (300K) is negligibly small. In this case, a 2J+1-fold degenerated primary multiplet split by an external crystal electric field can be treated as the basic contribution to the analysis of such systems' properties. In the case of approximate calculations for basis |J,Jz>, to determine which is the primary multiplet, the Hund principles, known from atomic physics, are applied:
The S, L and J of the ground multiplet are determined by Hund’s rules. The ground multiplet is 2J+1 degenerated – its degeneracy is removed by CEF interactions and magnetic interactions. CEF interactions and magnetic interactions resemble, somehow, Stark and Zeeman effect known from atomic physics. The energies and eigenfunctions of the discrete fine electronic structure are obtained by diagonalization of the (2J+1)-dimensional matrix. The fine electronic structure can be directly detected by many different spectroscopic methods. The one of the best of them is Inelastic Neutron Scattering (INS) experiments. The case of strong cubic CEF (for 3d transition -metal ions) interactions form group of levels (e.g. T2g, E.g., A2g) which are partially split by spin-orbit interactions and (if occur) lower symmetry CEF interactions. The energies and eigenfunctions of the discrete fine electronic structure (for the lowest term) are obtained by diagonalization of the (2L+1)(2S+1)-dimensional matrix. In the temperature T = 0[K] (absolute zero) only the lowest state is occupied. The magnetic moment at T=0[K] is equal to the moment of the ground state. It allows evaluate the total, spin and orbital moments. The eigenstates and corresponding eigenfunctions |Γn> can be found from direct diagonalization of hamiltonian matrix containing Crystal field and Spin–orbit interactions. Taking into consideration the thermal population of states, the thermal evolution of the single ion properties of the compound is established. This technique is based on the equivalent operator theory defined as the CEF widened by thermodynamic and analytical calculations defined as the supplement of the CEF theory by including thermodynamic and analytical calculations.
Examples of effective Hamiltonians
Hole bands of a bulk (3D) zinc-blende semiconductor will be split by
where
where the material parameter
where
Above expressions for spin–orbit interaction couple spin matrices
where
Electron spin in inhomogeneous magnetic field
Distinctive feature of spin-orbit interaction is presence in the Hamiltonian of a term that includes a product of orbital and spin operators. In atomic systems these are orbital and spin angular momenta
A similar effect can be achieved through the Larmor energy