Spin–orbit interaction

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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. 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.

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:

$\Delta H=-{\boldsymbol {\mu }}\cdot {\boldsymbol {B}},$

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^{[needs citation]}, there is one in the rest frame of the electron. Ignoring for now that this frame is not inertial, in SI units we end up with the equation

${\boldsymbol {B}}=-{{\boldsymbol {v}}\times {\boldsymbol {E}} \over c^{2}},$

where v is the velocity of the electron and E the electric field it travels through. Now we know that E is radial so we can rewrite ${\boldsymbol {E}}=\left|E/r\right|{\boldsymbol {r}}$ . Also we know that the momentum of the electron ${\boldsymbol {p}}=m_{{\text{e}}}{\boldsymbol {v}}$ . Substituting this in and changing the order of the cross product gives:

${\boldsymbol {B}}={{\boldsymbol {r}}\times {\boldsymbol {p}} \over m_{{\text{e}}}c^{2}}\left|{E \over r}\right|.$

Next, we express the electric field as the gradient of the electric potential ${\boldsymbol {E}}=-{\boldsymbol {\nabla }}V$ . Here we make the central field approximation, that is, that the electrostatic potential is spherically symmetric, so is only a function of radius. This approximation is exact for hydrogen, and indeed hydrogen-like systems. Now we can say

$\left|E\right|={\partial V \over \partial r}={1 \over e}{\partial U(r) \over \partial r},$

where $U=eV$ is the potential energy of the electron in the central field, and e is the elementary charge. Now we remember from classical mechanics that the angular momentum of a particle ${\boldsymbol {L}}={\boldsymbol {r}}\times {\boldsymbol {p}}$ . Putting it all together we get

${\boldsymbol {B}}={1 \over m_{{\text{e}}}ec^{2}}{1 \over r}{\partial U(r) \over \partial r}{\boldsymbol {L}}.$

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. (Recall that according to most conventions, the vector cross product yields a vector which is (often symbolically) oriented out of the plane of its two constituent vectors).

Magnetic moment of the electron

The magnetic moment of the electron is

${\boldsymbol {\mu }}_{S}=-g_{S}\mu _{B}{\frac {{\mathbf {S}}}{\hbar }}.$

where ${\boldsymbol {S}}$ is the spin angular momentum vector, $\mu _{{\text{B}}}$ is the Bohr magneton and $g_{{\text{s}}}\approx 2$ is the electron spin g-factor. Here, ${\boldsymbol {\mu }}$ is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum.

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

$\Delta H_{{\text{L}}}=-{\boldsymbol {\mu }}\cdot {\boldsymbol {B}}.$

Substituting in this equation expressions for the magnetic moment and the magnetic field, one gets

$\Delta H_{{\text{L}}}={2\mu _{{\text{B}}} \over \hbar m_{{\text{e}}}ec^{2}}{1 \over r}{\partial U(r) \over \partial r}{\boldsymbol {L}}\cdot {\boldsymbol {S}}.$

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.^[1] Thomas precession rate, ${\boldsymbol {\Omega }}_{{\text{T}}}$ , is related to the angular frequency of the orbital motion, ${\boldsymbol {\omega }}$ , of a spinning particle as follows ^[2]^[3]

${\boldsymbol {\Omega }}_{{\text{T}}}={\boldsymbol {\omega }}(\gamma -1),$

where $\gamma$ is the Lorentz factor of the moving particle. The Hamiltonian producing the spin precession ${\boldsymbol {\Omega }}_{{\text{T}}}$ is given by

$\Delta H_{{\text{T}}}={\boldsymbol {\Omega }}_{{\text{T}}}\cdot {\boldsymbol {S}}.$

To the first order in $(v/c)^{2}$ , we obtain

$\Delta H_{{\text{T}}}=-{\mu _{{\text{B}}} \over \hbar m_{{\text{e}}}ec^{2}}{1 \over r}{\partial U(r) \over \partial r}{\boldsymbol {L}}\cdot {\boldsymbol {S}}.$

Total interaction energy

The total spin–orbit potential in an external electrostatic potential takes the form

$\Delta H\equiv \Delta H_{{\text{L}}}+\Delta H_{{\text{T}}}={\mu _{{\text{B}}} \over \hbar m_{{\text{e}}}ec^{2}}{1 \over r}{\partial U(r) \over \partial r}{\boldsymbol {L}}\cdot {\boldsymbol {S}}.$

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

${\boldsymbol {J}}={\boldsymbol {L}}+{\boldsymbol {S}}.$

Taking the dot product of this with itself, we get

${\boldsymbol {J}}^{2}={\boldsymbol {L}}^{2}+{\boldsymbol {S}}^{2}+2{\boldsymbol {L}}\cdot {\boldsymbol {S}}$

(since L and S commute), and therefore

${\boldsymbol {L}}\cdot {\boldsymbol {S}}={1 \over 2}({\boldsymbol {J}}^{2}-{\boldsymbol {L}}^{2}-{\boldsymbol {S}}^{2})$

It can be shown that the five operators H₀, J², L², S², and J_z 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 j_z (the "z-component of total angular momentum").

To evaluate the energies, we note that

$\left\langle {1 \over r^{3}}\right\rangle ={\frac {2}{a^{3}n^{3}l(l+1)(2l+1)}}$

for hydrogenic wavefunctions (here $a=\hbar /Z\alpha m_{{\text{e}}}c$ is the Bohr radius divided by the nuclear charge Z); and

$\left\langle {\boldsymbol {L}}\cdot {\boldsymbol {S}}\right\rangle ={1 \over 2}(\langle {\boldsymbol {J}}^{2}\rangle -\langle {\boldsymbol {L}}^{2}\rangle -\langle {\boldsymbol {S}}^{2}\rangle )$

$={\hbar ^{2} \over 2}(j(j+1)-l(l+1)-s(s+1))$

Final energy shift

We can now say

$\Delta E={\beta \over 2}(j(j+1)-l(l+1)-s(s+1))$

where

$\beta =\beta (n,l)=Z^{4}{\mu _{0} \over 4{\pi }}g_{{\text{s}}}\mu _{{\text{B}}}^{2}{1 \over n^{3}a_{0}^{3}l(l+1/2)(l+1)}$

Spin–orbit interaction in solids

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 relatively more important role if we zoom in to bands close to the Fermi level ( $E_{{\text{F}}}$ ). The atomic ${\boldsymbol {L}}\cdot {\boldsymbol {S}}$ interaction for example splits bands which would be otherwise degenerate and the particular form of this spin–orbit splitting (typically of the order of few to few hundred millielectronvolts) depends on the particular system. The bands of interest can be then described by various effective models, usually based on some perturbative approach. An example of how the atomic spin–orbit interaction influences the band structure of a crystal is explained in the article about Rashba interaction.

Examples of effective Hamiltonians

Hole bands of a bulk (3D) zinc-blende semiconductor will be split by $\Delta _{0}$ into heavy and light holes (which form a $\Gamma _{8}$ quadruplet in the $\Gamma$ -point of the Brillouin zone) and a split-off band ( $\Gamma _{7}$ doublet). Including two conduction bands ( $\Gamma _{6}$ doublet in the $\Gamma$ -point), the system is described by the effective eight-band model of Kohn and Luttinger. If only top of the valence band is of interest (for example when $E_{{\text{F}}}\ll \Delta _{0}$ , Fermi level measured from the top of the valence band), the proper four-band effective model is

$H_{{\text{KL}}}(k_{{\text{x}}},k_{{\text{y}}},k_{{\text{z}}})=-{\frac {\hbar ^{2}}{2m}}\left[(\gamma _{1}+{\textstyle {\frac 52}\gamma _{2}})k^{2}-2\gamma _{2}(J_{{\text{x}}}^{2}k_{{\text{x}}}^{2}+J_{{\text{y}}}^{2}k_{{\text{y}}}^{2}+J_{{\text{z}}}^{2}k_{{\text{z}}}^{2})-2\gamma _{3}\sum _{{m\neq n}}J_{m}J_{n}k_{m}k_{n}\right]$

where $\gamma _{{1,2,3}}$ are the Luttinger parameters (analogous to the single effective mass of a one-band model of electrons) and $J_{{{\text{x}},{\text{y}},{\text{z}}}}$ are angular momentum 3/2 matrices ( $m$ is the free electron mass). In combination with magnetization, this type of spin–orbit interaction will distort the electronic bands depending on the magnetization direction, thereby causing Magnetocrystalline anisotropy (a special type of Magnetic anisotropy). If the semiconductor moreover lacks the inversion symmetry, the hole bands will exhibit cubic Dresselhaus splitting. Within the four bands (light and heavy holes), the dominant term is

$H_{{{\text{D}}3}}=b_{{41}}^{{8{\text{v}}8{\text{v}}}}[(k_{{\text{x}}}k_{{\text{y}}}^{2}-k_{{\text{x}}}k_{{\text{z}}}^{2})J_{{\text{x}}}+(k_{{\text{y}}}k_{{\text{z}}}^{2}-k_{{\text{y}}}k_{{\text{x}}}^{2})J_{{\text{y}}}+(k_{{\text{z}}}k_{{\text{x}}}^{2}-k_{{\text{z}}}k_{{\text{y}}}^{2})J_{{\text{z}}}]$

where the material parameter $b_{{41}}^{{8{\text{v}}8{\text{v}}}}=-81.93\,{\text{meV}}\cdot {\text{nm}}^{3}$ for GaAs (see pp. 72 in Winkler's book, according to more recent data the Dresselhaus constant in GaAs is 9 eVÅ³;^[4] the total Hamiltonian will be $H_{{\text{KL}}}+H_{{{\text{D}}3}}$ ). Two-dimensional electron gas in an asymmetric quantum well (or heterostructure) will feel the Rashba interaction. The appropriate two-band effective Hamiltonian is

$H_{0}+H_{{\text{R}}}={\frac {\hbar ^{2}k^{2}}{2m^{*}}}\sigma _{0}+\alpha (k_{{\text{y}}}\sigma _{{\text{x}}}-k_{{\text{x}}}\sigma _{{\text{y}}})$

where $\sigma _{0}$ is the 2 × 2 identity matrix, $\sigma _{{{\text{x}},{\text{y}}}}$ the Pauli matrices and $m^{*}$ the electron effective mass. The spin–orbit part of the Hamiltonian, $H_{{\text{R}}}$ is parametrized by $\alpha$ , sometimes called the Rashba parameter (its definition somewhat varies), which is related to the structure asymmetry.

Above expressions for spin–orbit interaction couple spin matrices ${\boldsymbol {J}}$ and ${\boldsymbol {\sigma }}$ to the quasi-momentum ${\boldsymbol {k}}$ , and to the vector potential ${\boldsymbol {A}}$ of an AC electric field through the Peierls substitution ${{\boldsymbol {k}}}=-i\nabla -({\frac {e}{\hbar c}}){{\boldsymbol {A}}}$ . They are lower order terms of the Luttinger–Kohn ${\boldsymbol {k}}\cdot {{\boldsymbol {p}}}$ expansion in powers of $k$ . Next terms of this expansion also produce terms that couple spin operators of the electron coordinate ${\boldsymbol {r}}$ . Indeed, a cross product $({\boldsymbol {\sigma }}\times {{\boldsymbol {k}}})$ is invariant with respect to time inversion. In cubit crystals, it has a symmetry of a vector and acquires a meaning of a spin–orbit contribution ${{\boldsymbol {r}}}_{{{\text{SO}}}}$ to the operator of coordinate. For electrons in semiconductors with a narrow gap $E_{G}$ between the conduction and heavy hole bands, Yafet derived the equation

${{\boldsymbol {r}}}_{{{\text{SO}}}}={\frac {\hbar ^{2}g}{4m_{0}}}\left({\frac {1}{E_{G}}}+{\frac {1}{E_{{G}}+\Delta _{0}}}\right)({\boldsymbol {\sigma }}\times {{\boldsymbol {k}}})$

where $m_{0}$ is a free electron mass, and $g$ is a $g$ -factor properly renormalized for spin–orbit interaction. This operator couples electron spin ${{\boldsymbol {S}}}={\frac {1}{2}}{{\boldsymbol {\sigma }}}$ directly to the electric field ${\boldsymbol {E}}$ through the interaction energy $-e({{\boldsymbol {r}}}_{{{\text{SO}}}}\cdot {{\boldsymbol {E}}})$ .

References

↑ L. H. Thomas, The motion of the spinning electron, Nature (London), 117, 514 (1926).
↑ L. Föppl and P. J. Daniell, Zur Kinematik des Born'schen starren Körpers, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 519 (1913).
↑ C. Møller, The Theory of Relativity, (Oxford at the Claredon Press, London, 1952).
↑ J. J. Krich and B. I. Halperin, Phys. Rev. Lett. 98, 226802 (2007)

Y. Yafet, in: {\it Solid State Physics}, ed. by F. Seitz and D. Turnbull (Academic, NY), v. {\bf 14}, p.1.

E. I. Rashba and V. I. Sheka, in: {\it Landau Level Spectroscopy} (North Holland, Amsterdam) 1991, p.131.

Textbooks

E. U. Condon and G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4.

D. J. Griffiths (2004). Introduction to Quantum Mechanics (2nd edition). Prentice Hall.

Landau, Lev; L. M. Lifshitz. " $\S$ 72. Fine structure of atomic levels". Quantum Mechanics: Non-Relativistic Theory, Volume 3. Cite uses deprecated parameters (help)

Yu, Peter Y.; Manuel Cardona. Fundamentals of Semiconductors. Cite uses deprecated parameters (help)

Winkler, Roland. Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems.

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