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Spin & Orbital moment

Spin and Charge Transport

Abstract:

In an electron gas the spins can be accumulated. In this case the spins of some delocalized conduction electrons are aligned in one direction. The delocalized electrons also usually have a non-zero orbital moment. The spin-orbit interaction is trying to align the spins along the direction of orbital moment. Therefore, a spin accumulation may be accompanied by an accumulation of the orbital moment. Therefore, it is not spin, but total magnetic moment is accumulated.

However, often the orbital moment of the delocalized electrons are quenched. It means the direction of the orbital moment in the atomic lattice is fixed and it can not be changed. Also, often the quenched orbital moment is well- decoupled from the electron spin. Therefore, it is not accumulation of the electron total magnetic moment, but separate accumulation of the spin and the orbital moment should be considered.

 


In short:

 

- The orbital moment of electrons of the conduction band is zero and the electrons of light- and heavy hole bands is 1.

- Because of the periodicity of the crystal lattice, the orbital of electrons (Bloch function) might not be able to rotate freely. In this case the orbital moment is piratically or fully quenched (frozen).

-In the case of the unquenched orbital moment, the orbital moment is parallel to the spin for electrons of the heavy-hole band and the orbital moment is opposite to the spin direction for electrons of the light-hole band. The different orientation of the orbital moment in the respect to the spin direction for heavy and light holes is the reason for the reduction of the magneto-optical response and the reduction of the efficiency for excitation of spin- polarized electrons by circular- polarized light.

 


How to know whether the orbital moment is quenched or not?

A. A good sign that the orbital moment is fully quenched is when the g-factor of electron is equal exactly 2 as in the case of a free electron in vacuum. The g-factor could be measured by the Electron Paramagnetic Resonance (EPR). Some researchers believe that difference of the g-factor from 2 means that the orbital moment is partiality unquenched. It is not correct. The g-factor may be significantly enlarged or be reduced due to the spin-orbit interaction (See here). The spin-orbit interaction is not related to the quenching of the orbital moment.


Orbital moment in an ordinary atomic gas and in a solid

Orbital moment in an ordinary atomic gas

click here or on picture to enlarge it

The green ellipses show the orbits of the atoms of the gas. The red arrows show the direction of the corresponded orbital moment of an atom. Each atom has a non-zero orbital moment and the time-inverse symmetry is broken for each atom. However, the orbitals moments are oriented randomly, therefore the time-inverse symmetry is not broken for the atomic gas as whole.

Key point: The time inverse-symmetry can be

For clearer understanding the orbitals is depicted as 2D ellipses.

 

In the gas the orbit of an atom can be in any direction as well as corresponded orbital moment. Even though the orbital moment of each individual may be non-zero and the time-inverse symmetry is broken for each atoms, for the atomic gas as whole the time-inverse symmetry is not broken, because of the random orientation of the orbitals moments for each atom in the atomic gas.

 

 

 

 

 

The precession of the orbital moment literally means the precession of electron orbit as well.

 

 

 

 

 

 

 

Precession of orbital moment of an atom of atomic gas around a magnetic field

click here or on picture to enlarge it

 

The green ellipses show the orbits of the atoms of the gas. The red arrows show the direction of the corresponded orbital moment of an atom.

The precession of the orbital moment literally means the precession of electron orbit as well.

For clearer understanding the orbitals is depicted as 2D ellipses.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Precession of orbital moment in a magnetic field

The precession of the orbital moment literally means the precession of electron orbit.

Green arrow shows direction of the orbital moment. Blue arrow shows the direction of the magnetic field

Elliptical orbital

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p-like orbital

 

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"frozen" or "quenched" orbital moment in the solid.

"Quenched" orbital moment in the solid

Why there can not be a precession of the orbital moment in a crystal?

The precession of the orbital moment literally means the precession of electron orbit.

Green arrow shows direction of the orbital moment. Blue arrow shows direction of the magnetic field

Electron orbitals in a solid

in crystal the electron orbitals interact with each other and this interaction defines the crystal structure. The electron orbitals can not move or rotates.

The electron orbitals are shown in pink, nuclears are shown in black.

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Imaginary case when an orbital moment of electrons in a crystal precess in a magnetic field

The electron orbits in crystal can not rotate. Otherwise, the bonding between neighbor atoms would alter and the crystal would collapse.

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The effect of "quenching" of the orbital moment is the known effect in magneto chemistry (See here).

 

 

 

 

 

 

 

 

 

 

 

 

 


 

Q. In the case when there is a spin accumulation in the electron gas, is there any accumulation of the orbital moment?

Short answer: Yes. There is an accumulation of the orbital moment.
Long answer:
When there is no spin accumulation and no external field, the time-inverse symmetry is not broken.
There is an equal amount of electrons with opposite direction of the orbital moment.
When there is a spin accumulation in the electron gas, it is always accompanied by an accumulation of the orbital moment. It is because the interaction between the spin and the orbital moment (It is not the spin-orbit interaction!) The accumulation of the orbital moment may be weak. For example, it is weak in the case of electrons of the conduction band in a semiconductor, where the orbital moment of the electrons is nearly zero. Another example of a weak accumulation of the orbital moment is the accumulation in a material with the quenched orbital moment.

Q. The accumulations of the spin and the orbital moments are of the same kind. Is it correct?

Short answer: No. The accumulations of the spins and orbital moment are of different kinds, but often they coexist in the electron gas at the same time.
Long answer
The electron spin in a crystal can be directed in any direction. The special distribution of an electron in a crystal does not change when its spin changes the direction. In the contrast, the orbital moment may have only fixed directions in the crystal. To each direction of the orbital moment, each specific electron distribution corresponds. In a crystal, there is a fixed number of possible different electron distributions and therefore possible direction and value of the orbital moment. The

 

 


 

 

Orbital moment & spacial symmetry & time-inverse symmetry

In an atomic gas the time-inverse symmetry may be broken for each individual atom. Since the random orientation of atoms in the gas, even though this is the case, the time-inverse symmetry for the atomic gas is not broken. The case of the solid is different. The electron orbitals are fixed in the space and the time-inverse symmetry should not be broken even locally.

In contrast, the spin direction of electrons in a solid is not fixed in the space. Therefore, locally the time-inverse symmetry can be broken for an electron state. It is a "spin" state, which has non-zero spin (See here) . Since the spins of "spin" states are directed randomly in all directions, the time-inverse symmetry is not broken for the electron gas as whole. Even though the electron gas contains the "spin" states and for each individual "spin" state the time-inverse symmetry is broken.

In a non-magnetic metal without a spin accumulation the time-inverse symmetry can not be broken even locally. Therefore, the Hamiltonian for the electron gas should not change for inverse time.

 

 

 

 

Luttinger Hamiltonian & orbital moment & symmetry

Band structure of a direct band semiconductor

 

The green ellipses show the orbits of the atoms of the gas. The red arrows show the direction of the corresponded orbital moment of an atom.

 

Band structure of semiconductor

Following explanations are based on the Luttinger Hamiltonian. The Luttinger Hamiltonian assumes that delocalized electrons in a solid have similar orbital moment as the free atoms of an atomic gas. More realistic Hamiltonian is the Luttinger-Kohn Hamiltonian, which is based on the symmetry of wave functions rather than the features of the orbital moment. Despite of some "rough" assumptions, the Luttinger Hamiltonian is still useful for understanding of features of delocalized electrons.

This explanation is based on an assumption that the orbital moment is not "frozen". S,L and J denote the spin, the orbital moment and the total moment. j denotes the projection of the total moment on chosen axis.

Conduction band:

s-like symmetry; each electron state can be occupied either by two or one electrons or be not occupied. The orbital moment is zero. The electron orbital is spherical.

Since the orbital is spherical, the conduction electrons should not experience the spin-orbit interaction. It is far from truth. The Bloch function of conduction electrons is not spherical, especially in the case of a compound semiconductor. The spin-orbit interaction can be huge for the conduction electrons. For example, in the case of GaAs the g-factor can be even negative for the conduction electrons.

Valence band:

p-like symmetry; each electron state can be occupied either by two or one electrons or be not occupied. There are 3 different bands: heavy holes, light holes and split-off (SO) band

split-off (SO) band

The total orbital momentum is zero. The electron orbital is spherical. Each electron state can be occupied either by two electrons of opposite spins or by one electron or be not occupied.

The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be

Since the orbital is spherical, the conduction electrons should not experience the spin-orbit interaction. According to the Luttinger-Kohn model the orbitals of electrons of SO band is not spherical.

 

Heavy-hole band

The total orbital momentum is one. Each electron state can be occupied either by two electrons of opposite spins and opposite orbital moments or by one electron or be not occupied.

Spin is directed along the orbital moment.

The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be

Light-hole band

The total orbital momentum is one. Each electron state can be occupied either by two electrons of opposite spins and opposite orbital moments or by one electron or be not occupied.

The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be

Spin is directed opposite to the orbital moment.

The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be

Luttinger Hamiltonian for light and heavy holes

from M.I. Dyakonov "Spin Physics in Semiconductors" Springer Series in Solid-State Sciences ISSN 0171-1873
The following proves that the orbital moment of heave holes is parallel to their spins and the orbital moment of light holes is opposite to their spins. Assumption of fully- unquenched orbital moment is used.

The time-inverse should not change the Hamiltonian. The quantum-mechanical momentum operator p changes sign with time reversal. Therefore, in the Hamiltonian it can be only in a product with other operator, which changes its sign with time reversal. If we assume that there is no any quenching of the orbital moment, due to LS-coupling (Russell–Saunders coupling) the total moment J should be used as such operator. In the Hamiltonian the time-inverse-symmetrical product should be included. In the case of a critical point (band min or max) at p=0, the Hamiltonian should be symmetrical for the reversal of the direction of p, then the product should be included into the Hamiltonian. The Hamiltonian, which describes both the heavy and light holes bands, should have 4 eigenvalues corresponded to the projections of the total moment -3/2; -1/2;1/2;3/2. Therefore, the Hamiltonian for the heavy and light holes bands can be expressed as

where A and B are constants and

 

Along the direction of electron movement (along the direction of p), the projections the total moment can be only -3/2; -1/2;1/2;3/2. Therefore

and

Solving Eq.(2.1), the energy of heavy holes (j=+/-3/2)can be calculated as

the energy of light holes (j=+/-1/2)can be calculated as

since B<0 , the effective electron mass for heavy holes is larger than for the light holes.


Orbital moment of electrons in a solid. Quenched or unquenched ???

 

Note: There is no "black or white" answer for this question. There are some effects, which are completely not influenced by the orbital moment of electrons (for example, EPR), but there are some effects, which are still influenced by the fully-quenched orbital moment (for example, the Zeeman effect for delocalized conduction electrons)
Note: Often it is difficult to separate between influences of spin-orbit interaction and partially- unquenched orbital moment on electron properties.

Delocalized conduction electrons.

The orbital moment is

(1) nearly-fully quenched in a mono-crystal metal

(2) nearly-fully unquenched in a amorphous metal

(3) partiality quenched in a poly-crystal metal

The effective length of delocalized electrons may be over 1000 lattice periods. Therefore, the periodicity of crystal strongly fixes the orbits of the delocalized electrons. In the crystal there is a strong force against any rotation or deformation of orbit of delocalized electron. Since the orbit can not be changed, it literally means that the orbital moment is "frozen" and it can not be changed.

Localized d-electrons.

The orbital moment

(1) nearly-fully quenched in a mono-crystal metal and a poly-crystal metal

(2) partially unquenched in a amorphous metal

 

 

Even though the d-electrons are localized, their size is substantial. There is a substantial exchange interaction between the d-electrons and delocalized conduction electrons and between neighbor d-electrons.

The d-electrons have t and e symmetrical states according to the ligand field theory. The energy separation between d and t state is called the ligand-field splitting parameter delta. Only in the case of a smaller delta, the d-electrons can be in the high spin state. It proves that there is a strong interaction between d-electrons and the lattice. The orbits of the d-electron are fixed by the lattice symmetry and it can not freely rotated. Definitely, the orbital moment of the d-electrons are quenched.

Localized f-electrons.

The orbital moment is only partially quenched

The f-electrons only weakly interact with neighbors and with delocalized conduction electrons.

Orbital moment & special symmetry

Each orbital moment corresponds to the specific symmetry of electron orbit (the symmetry of the Bloch function). Even though the orbital moment is quenched, still the electron orbit has the symmetry corresponded to the orbital moment. The symmetry often defines many properties of electrons.

For example, the symmetry of atomic orbitals (Bloch functions) of electrons of the split-off band (orbital moment L=0) is nearly spherical. In contrast, the symmetry of atomic orbitals of electrons of the light-hole and heavy-hole bands (orbital moment L=1) is only rotational around one axis. For this reason, the energy of electrons of split-off band is different (smaller) than the energy of heavy and light holes even at p=0.

 

 

 



 

 

 

 

 

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